Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvnprodlem1 Structured version   Visualization version   GIF version

Theorem dvnprodlem1 45477
Description: 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
dvnprodlem1.c 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))
dvnprodlem1.j (𝜑𝐽 ∈ ℕ0)
dvnprodlem1.d 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)
dvnprodlem1.t (𝜑𝑇 ∈ Fin)
dvnprodlem1.z (𝜑𝑍𝑇)
dvnprodlem1.zr (𝜑 → ¬ 𝑍𝑅)
dvnprodlem1.rzt (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
Assertion
Ref Expression
dvnprodlem1 (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
Distinct variable groups:   𝐶,𝑐,𝑘   𝐷,𝑐,𝑡   𝐽,𝑐,𝑘,𝑛,𝑡   𝑅,𝑐,𝑘,𝑛,𝑡   𝑅,𝑠,𝑐,𝑛,𝑡   𝑡,𝑇,𝑠   𝑍,𝑐,𝑘,𝑛,𝑡   𝑍,𝑠   𝜑,𝑐,𝑛,𝑡,𝑘   𝜑,𝑠
Allowed substitution hints:   𝐶(𝑡,𝑛,𝑠)   𝐷(𝑘,𝑛,𝑠)   𝑇(𝑘,𝑛,𝑐)   𝐽(𝑠)

Proof of Theorem dvnprodlem1
Dummy variables 𝑑 𝑒 𝑚 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2726 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)
2 0zd 12608 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ∈ ℤ)
3 dvnprodlem1.j . . . . . . . . . . . . 13 (𝜑𝐽 ∈ ℕ0)
43nn0zd 12622 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℤ)
54adantr 479 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℤ)
6 dvnprodlem1.c . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))
7 oveq2 7427 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m (𝑅 ∪ {𝑍})))
8 rabeq 3433 . . . . . . . . . . . . . . . . . . . . . . 23 (((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛})
97, 8syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛})
10 sumeq1 15679 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡𝑠 (𝑐𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡))
1110eqeq1d 2727 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡𝑠 (𝑐𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛))
1211rabbidv 3426 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛})
139, 12eqtrd 2765 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛})
1413mpteq2dv 5251 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = (𝑅 ∪ {𝑍}) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛}))
15 dvnprodlem1.rzt . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
16 dvnprodlem1.t . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑇 ∈ Fin)
17 ssexg 5324 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∪ {𝑍}) ⊆ 𝑇𝑇 ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ V)
1815, 16, 17syl2anc 582 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
19 elpwg 4607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
2018, 19syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
2115, 20mpbird 256 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
22 nn0ex 12516 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
2322mptex 7235 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛}) ∈ V
2423a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛}) ∈ V)
256, 14, 21, 24fvmptd3 7027 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛}))
26 oveq2 7427 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
2726oveq1d 7434 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝐽 → ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
28 rabeq 3433 . . . . . . . . . . . . . . . . . . . . . 22 (((0...𝑛) ↑m (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛})
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛})
30 eqeq2 2737 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽))
3130rabbidv 3426 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
3229, 31eqtrd 2765 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
3332adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 𝐽) → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
34 ovex 7452 . . . . . . . . . . . . . . . . . . . . 21 ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∈ V
3534rabex 5335 . . . . . . . . . . . . . . . . . . . 20 {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ∈ V
3635a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ∈ V)
3725, 33, 3, 36fvmptd 7011 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
38 ssrab2 4073 . . . . . . . . . . . . . . . . . . 19 {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍}))
3938a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
4037, 39eqsstrd 4015 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
4140adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
42 simpr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
4341, 42sseldd 3977 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
44 elmapi 8868 . . . . . . . . . . . . . . 15 (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
4543, 44syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
46 dvnprodlem1.z . . . . . . . . . . . . . . . . 17 (𝜑𝑍𝑇)
47 snidg 4664 . . . . . . . . . . . . . . . . 17 (𝑍𝑇𝑍 ∈ {𝑍})
4846, 47syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ {𝑍})
49 elun2 4175 . . . . . . . . . . . . . . . 16 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
5048, 49syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑍 ∈ (𝑅 ∪ {𝑍}))
5150adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
5245, 51ffvelcdmd 7094 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) ∈ (0...𝐽))
5352elfzelzd 13542 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) ∈ ℤ)
545, 53zsubcld 12709 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) ∈ ℤ)
55 elfzle2 13545 . . . . . . . . . . . . 13 ((𝑐𝑍) ∈ (0...𝐽) → (𝑐𝑍) ≤ 𝐽)
5652, 55syl 17 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) ≤ 𝐽)
575zred 12704 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℝ)
5853zred 12704 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) ∈ ℝ)
5957, 58subge0d 11841 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝐽 − (𝑐𝑍)) ↔ (𝑐𝑍) ≤ 𝐽))
6056, 59mpbird 256 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝐽 − (𝑐𝑍)))
61 elfzle1 13544 . . . . . . . . . . . . 13 ((𝑐𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐𝑍))
6252, 61syl 17 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝑐𝑍))
6357, 58subge02d 11843 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝑐𝑍) ↔ (𝐽 − (𝑐𝑍)) ≤ 𝐽))
6462, 63mpbid 231 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) ≤ 𝐽)
652, 5, 54, 60, 64elfzd 13532 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) ∈ (0...𝐽))
66 elmapfn 8884 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) → 𝑐 Fn (𝑅 ∪ {𝑍}))
6743, 66syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
68 ssun1 4170 . . . . . . . . . . . . . . . . . 18 𝑅 ⊆ (𝑅 ∪ {𝑍})
6968a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
70 fnssres 6679 . . . . . . . . . . . . . . . . 17 ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑅 ⊆ (𝑅 ∪ {𝑍})) → (𝑐𝑅) Fn 𝑅)
7167, 69, 70syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅) Fn 𝑅)
72 nfv 1909 . . . . . . . . . . . . . . . . . 18 𝑡𝜑
73 nfcv 2891 . . . . . . . . . . . . . . . . . . 19 𝑡𝑐
74 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡𝒫 𝑇
75 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡0
76 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡𝑠
7776nfsum1 15680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡Σ𝑡𝑠 (𝑐𝑡)
78 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡𝑛
7977, 78nfeq 2905 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡Σ𝑡𝑠 (𝑐𝑡) = 𝑛
80 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡((0...𝑛) ↑m 𝑠)
8179, 80nfrabw 3456 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡{𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}
8275, 81nfmpt 5256 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛})
8374, 82nfmpt 5256 . . . . . . . . . . . . . . . . . . . . . 22 𝑡(𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))
846, 83nfcxfr 2889 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝐶
85 nfcv 2891 . . . . . . . . . . . . . . . . . . . . 21 𝑡(𝑅 ∪ {𝑍})
8684, 85nffv 6906 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝐶‘(𝑅 ∪ {𝑍}))
87 nfcv 2891 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐽
8886, 87nffv 6906 . . . . . . . . . . . . . . . . . . 19 𝑡((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
8973, 88nfel 2906 . . . . . . . . . . . . . . . . . 18 𝑡 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
9072, 89nfan 1894 . . . . . . . . . . . . . . . . 17 𝑡(𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
91 fvres 6915 . . . . . . . . . . . . . . . . . . . 20 (𝑡𝑅 → ((𝑐𝑅)‘𝑡) = (𝑐𝑡))
9291adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → ((𝑐𝑅)‘𝑡) = (𝑐𝑡))
932adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 0 ∈ ℤ)
9454adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝐽 − (𝑐𝑍)) ∈ ℤ)
9545adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
9669sselda 3976 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
9795, 96ffvelcdmd 7094 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) ∈ (0...𝐽))
9897elfzelzd 13542 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) ∈ ℤ)
99 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐𝑡) ∈ (0...𝐽) → 0 ≤ (𝑐𝑡))
10097, 99syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 0 ≤ (𝑐𝑡))
10115unssad 4185 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑅𝑇)
102 ssfi 9201 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ Fin ∧ 𝑅𝑇) → 𝑅 ∈ Fin)
10316, 101, 102syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑅 ∈ Fin)
104103ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 𝑅 ∈ Fin)
105 fzssz 13543 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0...𝐽) ⊆ ℤ
106 zssre 12603 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ℤ ⊆ ℝ
107105, 106sstri 3986 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0...𝐽) ⊆ ℝ
108107a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → (0...𝐽) ⊆ ℝ)
10945adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
11069sselda 3976 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → 𝑟 ∈ (𝑅 ∪ {𝑍}))
111109, 110ffvelcdmd 7094 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → (𝑐𝑟) ∈ (0...𝐽))
112108, 111sseldd 3977 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → (𝑐𝑟) ∈ ℝ)
113112adantlr 713 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) ∧ 𝑟𝑅) → (𝑐𝑟) ∈ ℝ)
114 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐𝑟) ∈ (0...𝐽) → 0 ≤ (𝑐𝑟))
115111, 114syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → 0 ≤ (𝑐𝑟))
116115adantlr 713 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) ∧ 𝑟𝑅) → 0 ≤ (𝑐𝑟))
117 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 = 𝑡 → (𝑐𝑟) = (𝑐𝑡))
118 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → 𝑡𝑅)
119104, 113, 116, 117, 118fsumge1 15787 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) ≤ Σ𝑟𝑅 (𝑐𝑟))
120103adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
121112recnd 11279 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟𝑅) → (𝑐𝑟) ∈ ℂ)
122120, 121fsumcl 15723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟𝑅 (𝑐𝑟) ∈ ℂ)
12358recnd 11279 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) ∈ ℂ)
124122, 123pncand 11609 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟𝑅 (𝑐𝑟) + (𝑐𝑍)) − (𝑐𝑍)) = Σ𝑟𝑅 (𝑐𝑟))
125 nfv 1909 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑟(𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
126 nfcv 2891 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑟(𝑐𝑍)
12746adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍𝑇)
128 dvnprodlem1.zr . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ¬ 𝑍𝑅)
129128adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍𝑅)
130 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑟 = 𝑍 → (𝑐𝑟) = (𝑐𝑍))
131125, 126, 120, 127, 129, 121, 130, 123fsumsplitsn 15734 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐𝑟) = (Σ𝑟𝑅 (𝑐𝑟) + (𝑐𝑍)))
132131eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟𝑅 (𝑐𝑟) + (𝑐𝑍)) = Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐𝑟))
133117cbvsumv 15686 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡)
134133a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡))
13537adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
13642, 135eleqtrd 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
137 rabid 3439 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ↔ (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽))
138136, 137sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽))
139138simprd 494 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽)
140134, 139eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐𝑟) = 𝐽)
141132, 140eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟𝑅 (𝑐𝑟) + (𝑐𝑍)) = 𝐽)
142141oveq1d 7434 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟𝑅 (𝑐𝑟) + (𝑐𝑍)) − (𝑐𝑍)) = (𝐽 − (𝑐𝑍)))
143124, 142eqtr3d 2767 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟𝑅 (𝑐𝑟) = (𝐽 − (𝑐𝑍)))
144143adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → Σ𝑟𝑅 (𝑐𝑟) = (𝐽 − (𝑐𝑍)))
145119, 144breqtrd 5175 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) ≤ (𝐽 − (𝑐𝑍)))
14693, 94, 98, 100, 145elfzd 13532 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) ∈ (0...(𝐽 − (𝑐𝑍))))
14792, 146eqeltrd 2825 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → ((𝑐𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐𝑍))))
148147ex 411 . . . . . . . . . . . . . . . . 17 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡𝑅 → ((𝑐𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐𝑍)))))
14990, 148ralrimi 3244 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡𝑅 ((𝑐𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐𝑍))))
15071, 149jca 510 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐𝑅) Fn 𝑅 ∧ ∀𝑡𝑅 ((𝑐𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐𝑍)))))
151 ffnfv 7128 . . . . . . . . . . . . . . 15 ((𝑐𝑅):𝑅⟶(0...(𝐽 − (𝑐𝑍))) ↔ ((𝑐𝑅) Fn 𝑅 ∧ ∀𝑡𝑅 ((𝑐𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐𝑍)))))
152150, 151sylibr 233 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅):𝑅⟶(0...(𝐽 − (𝑐𝑍))))
153 ovexd 7454 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...(𝐽 − (𝑐𝑍))) ∈ V)
15416, 101ssexd 5325 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ V)
155154adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ V)
156153, 155elmapd 8859 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐𝑅) ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ↔ (𝑐𝑅):𝑅⟶(0...(𝐽 − (𝑐𝑍)))))
157152, 156mpbird 256 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅) ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅))
15891a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡𝑅 → ((𝑐𝑅)‘𝑡) = (𝑐𝑡)))
15990, 158ralrimi 3244 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡𝑅 ((𝑐𝑅)‘𝑡) = (𝑐𝑡))
160159sumeq2d 15692 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡𝑅 ((𝑐𝑅)‘𝑡) = Σ𝑡𝑅 (𝑐𝑡))
161117cbvsumv 15686 . . . . . . . . . . . . . . . 16 Σ𝑟𝑅 (𝑐𝑟) = Σ𝑡𝑅 (𝑐𝑡)
162161eqcomi 2734 . . . . . . . . . . . . . . 15 Σ𝑡𝑅 (𝑐𝑡) = Σ𝑟𝑅 (𝑐𝑟)
163162a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡𝑅 (𝑐𝑡) = Σ𝑟𝑅 (𝑐𝑟))
164143idi 1 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟𝑅 (𝑐𝑟) = (𝐽 − (𝑐𝑍)))
165160, 163, 1643eqtrd 2769 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡𝑅 ((𝑐𝑅)‘𝑡) = (𝐽 − (𝑐𝑍)))
166157, 165jca 510 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐𝑅) ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∧ Σ𝑡𝑅 ((𝑐𝑅)‘𝑡) = (𝐽 − (𝑐𝑍))))
167 eqidd 2726 . . . . . . . . . . . . . . 15 (𝑒 = (𝑐𝑅) → 𝑅 = 𝑅)
168 simpl 481 . . . . . . . . . . . . . . . 16 ((𝑒 = (𝑐𝑅) ∧ 𝑡𝑅) → 𝑒 = (𝑐𝑅))
169168fveq1d 6898 . . . . . . . . . . . . . . 15 ((𝑒 = (𝑐𝑅) ∧ 𝑡𝑅) → (𝑒𝑡) = ((𝑐𝑅)‘𝑡))
170167, 169sumeq12rdv 15697 . . . . . . . . . . . . . 14 (𝑒 = (𝑐𝑅) → Σ𝑡𝑅 (𝑒𝑡) = Σ𝑡𝑅 ((𝑐𝑅)‘𝑡))
171170eqeq1d 2727 . . . . . . . . . . . . 13 (𝑒 = (𝑐𝑅) → (Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍)) ↔ Σ𝑡𝑅 ((𝑐𝑅)‘𝑡) = (𝐽 − (𝑐𝑍))))
172171elrab 3679 . . . . . . . . . . . 12 ((𝑐𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))} ↔ ((𝑐𝑅) ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∧ Σ𝑡𝑅 ((𝑐𝑅)‘𝑡) = (𝐽 − (𝑐𝑍))))
173166, 172sylibr 233 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))})
174 oveq2 7427 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑅 → ((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m 𝑅))
175 rabeq 3433 . . . . . . . . . . . . . . . . . . 19 (((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m 𝑅) → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛})
176174, 175syl 17 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛})
177 sumeq1 15679 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑅 → Σ𝑡𝑠 (𝑐𝑡) = Σ𝑡𝑅 (𝑐𝑡))
178177eqeq1d 2727 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑅 → (Σ𝑡𝑠 (𝑐𝑡) = 𝑛 ↔ Σ𝑡𝑅 (𝑐𝑡) = 𝑛))
179178rabbidv 3426 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
180176, 179eqtrd 2765 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
181180mpteq2dv 5251 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑅 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}))
182 elpwg 4607 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇𝑅𝑇))
183154, 182syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅 ∈ 𝒫 𝑇𝑅𝑇))
184101, 183mpbird 256 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ 𝒫 𝑇)
18522mptex 7235 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}) ∈ V
186185a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}) ∈ V)
1876, 181, 184, 186fvmptd3 7027 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}))
188187adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}))
189 oveq2 7427 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
190189oveq1d 7434 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → ((0...𝑛) ↑m 𝑅) = ((0...𝑚) ↑m 𝑅))
191 rabeq 3433 . . . . . . . . . . . . . . . . . 18 (((0...𝑛) ↑m 𝑅) = ((0...𝑚) ↑m 𝑅) → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
192190, 191syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
193 eqeq2 2737 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (Σ𝑡𝑅 (𝑐𝑡) = 𝑛 ↔ Σ𝑡𝑅 (𝑐𝑡) = 𝑚))
194193rabbidv 3426 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚})
195192, 194eqtrd 2765 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚})
196195cbvmptv 5262 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚})
197196a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚}))
198188, 197eqtrd 2765 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶𝑅) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚}))
199 fveq1 6895 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑒 → (𝑐𝑡) = (𝑒𝑡))
200199sumeq2sdv 15694 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑒 → Σ𝑡𝑅 (𝑐𝑡) = Σ𝑡𝑅 (𝑒𝑡))
201200eqeq1d 2727 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑒 → (Σ𝑡𝑅 (𝑐𝑡) = 𝑚 ↔ Σ𝑡𝑅 (𝑒𝑡) = 𝑚))
202201cbvrabv 3429 . . . . . . . . . . . . . . . 16 {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚}
203202a1i 11 . . . . . . . . . . . . . . 15 (𝑚 = (𝐽 − (𝑐𝑍)) → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚})
204 oveq2 7427 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝐽 − (𝑐𝑍)) → (0...𝑚) = (0...(𝐽 − (𝑐𝑍))))
205204oveq1d 7434 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐽 − (𝑐𝑍)) → ((0...𝑚) ↑m 𝑅) = ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅))
206 rabeq 3433 . . . . . . . . . . . . . . . 16 (((0...𝑚) ↑m 𝑅) = ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) → {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚})
207205, 206syl 17 . . . . . . . . . . . . . . 15 (𝑚 = (𝐽 − (𝑐𝑍)) → {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚})
208 eqeq2 2737 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐽 − (𝑐𝑍)) → (Σ𝑡𝑅 (𝑒𝑡) = 𝑚 ↔ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))))
209208rabbidv 3426 . . . . . . . . . . . . . . 15 (𝑚 = (𝐽 − (𝑐𝑍)) → {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))})
210203, 207, 2093eqtrd 2769 . . . . . . . . . . . . . 14 (𝑚 = (𝐽 − (𝑐𝑍)) → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))})
211210adantl 480 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑚 = (𝐽 − (𝑐𝑍))) → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))})
21254, 60jca 510 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐𝑍))))
213 elnn0z 12609 . . . . . . . . . . . . . 14 ((𝐽 − (𝑐𝑍)) ∈ ℕ0 ↔ ((𝐽 − (𝑐𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐𝑍))))
214212, 213sylibr 233 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) ∈ ℕ0)
215 ovex 7452 . . . . . . . . . . . . . . 15 ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∈ V
216215rabex 5335 . . . . . . . . . . . . . 14 {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))} ∈ V
217216a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))} ∈ V)
218198, 211, 214, 217fvmptd 7011 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))) = {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))})
219218eqcomd 2731 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐𝑍))) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑒𝑡) = (𝐽 − (𝑐𝑍))} = ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))
220173, 219eleqtrd 2827 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))
22165, 220jca 510 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍)))))
2221, 221jca 510 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∧ ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))))
223 ovexd 7454 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) ∈ V)
224 vex 3465 . . . . . . . . . . 11 𝑐 ∈ V
225224resex 6034 . . . . . . . . . 10 (𝑐𝑅) ∈ V
226225a1i 11 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑅) ∈ V)
227 opeq12 4877 . . . . . . . . . . . 12 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → ⟨𝑘, 𝑑⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)
228227eqeq2d 2736 . . . . . . . . . . 11 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → (⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ↔ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩))
229 eleq1 2813 . . . . . . . . . . . . 13 (𝑘 = (𝐽 − (𝑐𝑍)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐𝑍)) ∈ (0...𝐽)))
230229adantr 479 . . . . . . . . . . . 12 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐𝑍)) ∈ (0...𝐽)))
231 simpr 483 . . . . . . . . . . . . 13 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → 𝑑 = (𝑐𝑅))
232 fveq2 6896 . . . . . . . . . . . . . 14 (𝑘 = (𝐽 − (𝑐𝑍)) → ((𝐶𝑅)‘𝑘) = ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))
233232adantr 479 . . . . . . . . . . . . 13 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → ((𝐶𝑅)‘𝑘) = ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))
234231, 233eleq12d 2819 . . . . . . . . . . . 12 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → (𝑑 ∈ ((𝐶𝑅)‘𝑘) ↔ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍)))))
235230, 234anbi12d 630 . . . . . . . . . . 11 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → ((𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘)) ↔ ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))))
236228, 235anbi12d 630 . . . . . . . . . 10 ((𝑘 = (𝐽 − (𝑐𝑍)) ∧ 𝑑 = (𝑐𝑅)) → ((⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘))) ↔ (⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∧ ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍)))))))
237236spc2egv 3583 . . . . . . . . 9 (((𝐽 − (𝑐𝑍)) ∈ V ∧ (𝑐𝑅) ∈ V) → ((⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∧ ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))) → ∃𝑘𝑑(⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘)))))
238223, 226, 237syl2anc 582 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∧ ((𝐽 − (𝑐𝑍)) ∈ (0...𝐽) ∧ (𝑐𝑅) ∈ ((𝐶𝑅)‘(𝐽 − (𝑐𝑍))))) → ∃𝑘𝑑(⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘)))))
239222, 238mpd 15 . . . . . . 7 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∃𝑘𝑑(⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘))))
240 eliunxp 5840 . . . . . . 7 (⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∈ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ↔ ∃𝑘𝑑(⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶𝑅)‘𝑘))))
241239, 240sylibr 233 . . . . . 6 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∈ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
242 dvnprodlem1.d . . . . . 6 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)
243241, 242fmptd 7123 . . . . 5 (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
24488nfcri 2882 . . . . . . . . . . . 12 𝑡 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
24589, 244nfan 1894 . . . . . . . . . . 11 𝑡(𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
24672, 245nfan 1894 . . . . . . . . . 10 𝑡(𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
247 nfv 1909 . . . . . . . . . 10 𝑡(𝐷𝑐) = (𝐷𝑒)
248246, 247nfan 1894 . . . . . . . . 9 𝑡((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒))
24992eqcomd 2731 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡𝑅) → (𝑐𝑡) = ((𝑐𝑅)‘𝑡))
250249adantlrr 719 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ 𝑡𝑅) → (𝑐𝑡) = ((𝑐𝑅)‘𝑡))
251250adantlr 713 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → (𝑐𝑡) = ((𝑐𝑅)‘𝑡))
252242a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩))
253 opex 5466 . . . . . . . . . . . . . . . . . . . 20 ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∈ V
254253a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ ∈ V)
255252, 254fvmpt2d 7017 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷𝑐) = ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)
256255fveq2d 6900 . . . . . . . . . . . . . . . . 17 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (2nd ‘(𝐷𝑐)) = (2nd ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩))
257256fveq1d 6898 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2nd ‘(𝐷𝑐))‘𝑡) = ((2nd ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)‘𝑡))
258 ovex 7452 . . . . . . . . . . . . . . . . . . 19 (𝐽 − (𝑐𝑍)) ∈ V
259258, 225op2nd 8003 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩) = (𝑐𝑅)
260259fveq1i 6897 . . . . . . . . . . . . . . . . 17 ((2nd ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)‘𝑡) = ((𝑐𝑅)‘𝑡)
261260a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2nd ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)‘𝑡) = ((𝑐𝑅)‘𝑡))
262257, 261eqtr2d 2766 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐𝑅)‘𝑡) = ((2nd ‘(𝐷𝑐))‘𝑡))
263262adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝑐𝑅)‘𝑡) = ((2nd ‘(𝐷𝑐))‘𝑡))
264263ad2antrr 724 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → ((𝑐𝑅)‘𝑡) = ((2nd ‘(𝐷𝑐))‘𝑡))
265 simpr 483 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐷𝑐) = (𝐷𝑒))
266 fveq1 6895 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑒 → (𝑐𝑍) = (𝑒𝑍))
267266oveq2d 7435 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = 𝑒 → (𝐽 − (𝑐𝑍)) = (𝐽 − (𝑒𝑍)))
268 reseq1 5979 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = 𝑒 → (𝑐𝑅) = (𝑒𝑅))
269267, 268opeq12d 4883 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑒 → ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩)
270 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
271 opex 5466 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩ ∈ V
272271a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩ ∈ V)
273242, 269, 270, 272fvmptd3 7027 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷𝑒) = ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩)
274273adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐷𝑒) = ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩)
275265, 274eqtrd 2765 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐷𝑐) = ⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩)
276275fveq2d 6900 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (2nd ‘(𝐷𝑐)) = (2nd ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩))
277276adantlrl 718 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (2nd ‘(𝐷𝑐)) = (2nd ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩))
278277adantr 479 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → (2nd ‘(𝐷𝑐)) = (2nd ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩))
279 ovex 7452 . . . . . . . . . . . . . . . . . 18 (𝐽 − (𝑒𝑍)) ∈ V
280 vex 3465 . . . . . . . . . . . . . . . . . . 19 𝑒 ∈ V
281280resex 6034 . . . . . . . . . . . . . . . . . 18 (𝑒𝑅) ∈ V
282279, 281op2nd 8003 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩) = (𝑒𝑅)
283282a1i 11 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → (2nd ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩) = (𝑒𝑅))
284278, 283eqtrd 2765 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → (2nd ‘(𝐷𝑐)) = (𝑒𝑅))
285284fveq1d 6898 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → ((2nd ‘(𝐷𝑐))‘𝑡) = ((𝑒𝑅)‘𝑡))
286 fvres 6915 . . . . . . . . . . . . . . 15 (𝑡𝑅 → ((𝑒𝑅)‘𝑡) = (𝑒𝑡))
287286adantl 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → ((𝑒𝑅)‘𝑡) = (𝑒𝑡))
288285, 287eqtrd 2765 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → ((2nd ‘(𝐷𝑐))‘𝑡) = (𝑒𝑡))
289251, 264, 2883eqtrd 2769 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡𝑅) → (𝑐𝑡) = (𝑒𝑡))
290289adantlr 713 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → (𝑐𝑡) = (𝑒𝑡))
291 simpl 481 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
292 elunnel1 4146 . . . . . . . . . . . . . 14 ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡𝑅) → 𝑡 ∈ {𝑍})
293 elsni 4647 . . . . . . . . . . . . . 14 (𝑡 ∈ {𝑍} → 𝑡 = 𝑍)
294292, 293syl 17 . . . . . . . . . . . . 13 ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡𝑅) → 𝑡 = 𝑍)
295294adantll 712 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → 𝑡 = 𝑍)
296 simpr 483 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍)
297296fveq2d 6900 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 = 𝑍) → (𝑐𝑡) = (𝑐𝑍))
2983nn0cnd 12572 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ ℂ)
299298adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
300299, 123nncand 11613 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐𝑍))) = (𝑐𝑍))
301300eqcomd 2731 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐𝑍) = (𝐽 − (𝐽 − (𝑐𝑍))))
302301adantrr 715 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝑐𝑍) = (𝐽 − (𝐽 − (𝑐𝑍))))
303302adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝑐𝑍) = (𝐽 − (𝐽 − (𝑐𝑍))))
304255fveq2d 6900 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷𝑐)) = (1st ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩))
305258, 225op1st 8002 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩) = (𝐽 − (𝑐𝑍))
306305a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩) = (𝐽 − (𝑐𝑍)))
307304, 306eqtr2d 2766 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐𝑍)) = (1st ‘(𝐷𝑐)))
308307oveq2d 7435 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐𝑍))) = (𝐽 − (1st ‘(𝐷𝑐))))
309308adantrr 715 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝐽 − (𝐽 − (𝑐𝑍))) = (𝐽 − (1st ‘(𝐷𝑐))))
310309adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐽 − (𝐽 − (𝑐𝑍))) = (𝐽 − (1st ‘(𝐷𝑐))))
311 fveq2 6896 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷𝑐) = (𝐷𝑒) → (1st ‘(𝐷𝑐)) = (1st ‘(𝐷𝑒)))
312311adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (1st ‘(𝐷𝑐)) = (1st ‘(𝐷𝑒)))
313273fveq2d 6900 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷𝑒)) = (1st ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩))
314313adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (1st ‘(𝐷𝑒)) = (1st ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩))
315279, 281op1st 8002 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩) = (𝐽 − (𝑒𝑍))
316315a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (1st ‘⟨(𝐽 − (𝑒𝑍)), (𝑒𝑅)⟩) = (𝐽 − (𝑒𝑍)))
317312, 314, 3163eqtrd 2769 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (1st ‘(𝐷𝑐)) = (𝐽 − (𝑒𝑍)))
318317oveq2d 7435 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐽 − (1st ‘(𝐷𝑐))) = (𝐽 − (𝐽 − (𝑒𝑍))))
319298adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
320 zsscn 12604 . . . . . . . . . . . . . . . . . . . . . . 23 ℤ ⊆ ℂ
321105, 320sstri 3986 . . . . . . . . . . . . . . . . . . . . . 22 (0...𝐽) ⊆ ℂ
322321a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
323 eleq1w 2808 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑒 → (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↔ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
324323anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑒 → ((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ↔ (𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))))
325 feq1 6704 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑒 → (𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽) ↔ 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)))
326324, 325imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = 𝑒 → (((𝜑𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) ↔ ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))))
327326, 45chvarvv 1994 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))
32850adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
329327, 328ffvelcdmd 7094 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒𝑍) ∈ (0...𝐽))
330322, 329sseldd 3977 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒𝑍) ∈ ℂ)
331319, 330nncand 11613 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑒𝑍))) = (𝑒𝑍))
332331adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐽 − (𝐽 − (𝑒𝑍))) = (𝑒𝑍))
333 eqidd 2726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝑒𝑍) = (𝑒𝑍))
334318, 332, 3333eqtrd 2769 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐽 − (1st ‘(𝐷𝑐))) = (𝑒𝑍))
335334adantlrl 718 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝐽 − (1st ‘(𝐷𝑐))) = (𝑒𝑍))
336303, 310, 3353eqtrd 2769 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝑐𝑍) = (𝑒𝑍))
337336adantr 479 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 = 𝑍) → (𝑐𝑍) = (𝑒𝑍))
338 fveq2 6896 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑍 → (𝑒𝑡) = (𝑒𝑍))
339338eqcomd 2731 . . . . . . . . . . . . . . 15 (𝑡 = 𝑍 → (𝑒𝑍) = (𝑒𝑡))
340339adantl 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 = 𝑍) → (𝑒𝑍) = (𝑒𝑡))
341297, 337, 3403eqtrd 2769 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 = 𝑍) → (𝑐𝑡) = (𝑒𝑡))
342341adantlr 713 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → (𝑐𝑡) = (𝑒𝑡))
343291, 295, 342syl2anc 582 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → (𝑐𝑡) = (𝑒𝑡))
344290, 343pm2.61dan 811 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐𝑡) = (𝑒𝑡))
345344ex 411 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → (𝑐𝑡) = (𝑒𝑡)))
346248, 345ralrimi 3244 . . . . . . . 8 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = (𝑒𝑡))
34767adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑐 Fn (𝑅 ∪ {𝑍}))
348347adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
349327ffnd 6724 . . . . . . . . . . 11 ((𝜑𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
350349adantrl 714 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑒 Fn (𝑅 ∪ {𝑍}))
351350adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
352 eqfnfv 7039 . . . . . . . . 9 ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑒 Fn (𝑅 ∪ {𝑍})) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = (𝑒𝑡)))
353348, 351, 352syl2anc 582 . . . . . . . 8 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = (𝑒𝑡)))
354346, 353mpbird 256 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷𝑐) = (𝐷𝑒)) → 𝑐 = 𝑒)
355354ex 411 . . . . . 6 ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝐷𝑐) = (𝐷𝑒) → 𝑐 = 𝑒))
356355ralrimivva 3190 . . . . 5 (𝜑 → ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷𝑐) = (𝐷𝑒) → 𝑐 = 𝑒))
357243, 356jca 510 . . . 4 (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷𝑐) = (𝐷𝑒) → 𝑐 = 𝑒)))
358 dff13 7265 . . . 4 (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷𝑐) = (𝐷𝑒) → 𝑐 = 𝑒)))
359357, 358sylibr 233 . . 3 (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
360 eliun 5001 . . . . . . . . . . 11 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)))
361360biimpi 215 . . . . . . . . . 10 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)))
362361adantl 480 . . . . . . . . 9 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)))
363 nfv 1909 . . . . . . . . . . 11 𝑘𝜑
364 nfcv 2891 . . . . . . . . . . . 12 𝑘𝑝
365 nfiu1 5031 . . . . . . . . . . . 12 𝑘 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))
366364, 365nfel 2906 . . . . . . . . . . 11 𝑘 𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))
367363, 366nfan 1894 . . . . . . . . . 10 𝑘(𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
368 nfv 1909 . . . . . . . . . 10 𝑘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽}
369 nfv 1909 . . . . . . . . . . . . . . . . 17 𝑡 𝑘 ∈ (0...𝐽)
370 nfcv 2891 . . . . . . . . . . . . . . . . . 18 𝑡𝑝
371 nfcv 2891 . . . . . . . . . . . . . . . . . . 19 𝑡{𝑘}
372 nfcv 2891 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑅
37384, 372nffv 6906 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝐶𝑅)
374 nfcv 2891 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑘
375373, 374nffv 6906 . . . . . . . . . . . . . . . . . . 19 𝑡((𝐶𝑅)‘𝑘)
376371, 375nfxp 5711 . . . . . . . . . . . . . . . . . 18 𝑡({𝑘} × ((𝐶𝑅)‘𝑘))
377370, 376nfel 2906 . . . . . . . . . . . . . . . . 17 𝑡 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))
37872, 369, 377nf3an 1896 . . . . . . . . . . . . . . . 16 𝑡(𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)))
379 0zd 12608 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ∈ ℤ)
3804adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
3813803ad2antl1 1182 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
382 iftrue 4536 . . . . . . . . . . . . . . . . . . . 20 (𝑡𝑅 → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = ((2nd𝑝)‘𝑡))
383382adantl 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = ((2nd𝑝)‘𝑡))
384 simp1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝜑)
385 simp2 1134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
386 xp2nd 8027 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (2nd𝑝) ∈ ((𝐶𝑅)‘𝑘))
3873863ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (2nd𝑝) ∈ ((𝐶𝑅)‘𝑘))
388187adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘 ∈ (0...𝐽)) → (𝐶𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛}))
389 oveq2 7427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
390389oveq1d 7434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑘 → ((0...𝑛) ↑m 𝑅) = ((0...𝑘) ↑m 𝑅))
391 rabeq 3433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((0...𝑛) ↑m 𝑅) = ((0...𝑘) ↑m 𝑅) → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
392390, 391syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛})
393 eqeq2 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑘 → (Σ𝑡𝑅 (𝑐𝑡) = 𝑛 ↔ Σ𝑡𝑅 (𝑐𝑡) = 𝑘))
394393rabbidv 3426 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
395392, 394eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
396395adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑘 ∈ (0...𝐽)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
397 elfznn0 13634 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ0)
398397adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ0)
399 ovex 7452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0...𝑘) ↑m 𝑅) ∈ V
400399rabex 5335 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘} ∈ V
401400a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘} ∈ V)
402388, 396, 398, 401fvmptd 7011 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐶𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
4034023adant3 1129 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ((𝐶𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
404387, 403eleqtrd 2827 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (2nd𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘})
405 elrabi 3673 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘} → (2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅))
4064053ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽) ∧ (2nd𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘}) → (2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅))
407384, 385, 404, 406syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅))
408 elmapi 8868 . . . . . . . . . . . . . . . . . . . . . . 23 ((2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅) → (2nd𝑝):𝑅⟶(0...𝑘))
409407, 408syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (2nd𝑝):𝑅⟶(0...𝑘))
410409adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (2nd𝑝):𝑅⟶(0...𝑘))
411410ffvelcdmda 7093 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ∈ (0...𝑘))
412411elfzelzd 13542 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ∈ ℤ)
413383, 412eqeltrd 2825 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℤ)
414 simpl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
415294adantll 712 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → 𝑡 = 𝑍)
416 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑡 = 𝑍) → 𝑡 = 𝑍)
417128adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑡 = 𝑍) → ¬ 𝑍𝑅)
418416, 417eqneltrd 2845 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑡 = 𝑍) → ¬ 𝑡𝑅)
419418iffalsed 4541 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
4204193ad2antl1 1182 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
4214adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑡 = 𝑍) → 𝐽 ∈ ℤ)
4224213ad2antl1 1182 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ)
423 xp1st 8026 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ {𝑘})
424 elsni 4647 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st𝑝) ∈ {𝑘} → (1st𝑝) = 𝑘)
425423, 424syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) = 𝑘)
426425adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (1st𝑝) = 𝑘)
427105sseli 3972 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
428427adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑘 ∈ ℤ)
429426, 428eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (1st𝑝) ∈ ℤ)
4304293adant1 1127 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (1st𝑝) ∈ ℤ)
431430adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (1st𝑝) ∈ ℤ)
432422, 431zsubcld 12709 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st𝑝)) ∈ ℤ)
433420, 432eqeltrd 2825 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℤ)
434433adantlr 713 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℤ)
435414, 415, 434syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℤ)
436413, 435pm2.61dan 811 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℤ)
437409ffvelcdmda 7093 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ∈ (0...𝑘))
438 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑝)‘𝑡) ∈ (0...𝑘) → 0 ≤ ((2nd𝑝)‘𝑡))
439437, 438syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → 0 ≤ ((2nd𝑝)‘𝑡))
440382eqcomd 2731 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝑅 → ((2nd𝑝)‘𝑡) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
441440adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
442439, 441breqtrd 5175 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → 0 ≤ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
443442adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → 0 ≤ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
444 simpll 765 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → (𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))))
445 elfzle2 13545 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘𝐽)
446 elfzel2 13539 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
447446zred 12704 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
448107sseli 3972 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
449447, 448subge0d 11841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽𝑘) ↔ 𝑘𝐽))
450445, 449mpbird 256 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽𝑘))
451450adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (0...𝐽) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽𝑘))
4524513ad2antl2 1183 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽𝑘))
453384, 418sylan 578 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → ¬ 𝑡𝑅)
454453iffalsed 4541 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
4554263adant1 1127 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (1st𝑝) = 𝑘)
456455oveq2d 7435 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝐽 − (1st𝑝)) = (𝐽𝑘))
457456adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st𝑝)) = (𝐽𝑘))
458454, 457eqtr2d 2766 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽𝑘) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
459452, 458breqtrd 5175 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
460444, 415, 459syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → 0 ≤ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
461443, 460pm2.61dan 811 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ≤ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
462 simpl2 1189 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → 𝑘 ∈ (0...𝐽))
463 fzssz 13543 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0...𝑘) ⊆ ℤ
464463sseli 3972 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd𝑝)‘𝑡) ∈ ℤ)
465464zred 12704 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd𝑝)‘𝑡) ∈ ℝ)
466465adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd𝑝)‘𝑡) ∈ ℝ)
467448adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
468447adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
469 elfzle2 13545 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd𝑝)‘𝑡) ≤ 𝑘)
470469adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd𝑝)‘𝑡) ≤ 𝑘)
471445adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘𝐽)
472466, 467, 468, 470, 471letrd 11408 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd𝑝)‘𝑡) ≤ 𝐽)
473437, 462, 472syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ≤ 𝐽)
474473adantlr 713 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ≤ 𝐽)
475383, 474eqbrtrd 5171 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ≤ 𝐽)
476458eqcomd 2731 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = (𝐽𝑘))
477398nn0ge0d 12573 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
478447adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
479448adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
480478, 479subge02d 11843 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝐽)) → (0 ≤ 𝑘 ↔ (𝐽𝑘) ≤ 𝐽))
481477, 480mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → (𝐽𝑘) ≤ 𝐽)
482481adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝐽)) ∧ 𝑡 = 𝑍) → (𝐽𝑘) ≤ 𝐽)
4834823adantl3 1165 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽𝑘) ≤ 𝐽)
484476, 483eqbrtrd 5171 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ≤ 𝐽)
485444, 415, 484syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ≤ 𝐽)
486475, 485pm2.61dan 811 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ≤ 𝐽)
487379, 381, 436, 461, 486elfzd 13532 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ (0...𝐽))
488 eqid 2725 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
489378, 487, 488fmptdf 7126 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽))
490 ovexd 7454 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (0...𝐽) ∈ V)
491384, 18syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑅 ∪ {𝑍}) ∈ V)
492490, 491elmapd 8859 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ↔ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽)))
493489, 492mpbird 256 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})))
494 eqidd 2726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))))
495 eleq1w 2808 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝑡 → (𝑟𝑅𝑡𝑅))
496 fveq2 6896 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝑡 → ((2nd𝑝)‘𝑟) = ((2nd𝑝)‘𝑡))
497495, 496ifbieq1d 4554 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑡 → if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
498497adantl 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑟 = 𝑡) → if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
499 simpr 483 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
500494, 498, 499, 436fvmptd 7011 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
501500ex 411 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))))
502378, 501ralrimi 3244 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
503502sumeq2d 15692 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))
504 nfcv 2891 . . . . . . . . . . . . . . . 16 𝑡if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝)))
505384, 103syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑅 ∈ Fin)
506384, 46syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑍𝑇)
507384, 128syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ¬ 𝑍𝑅)
508382adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = ((2nd𝑝)‘𝑡))
509437elfzelzd 13542 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ∈ ℤ)
510509zcnd 12705 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → ((2nd𝑝)‘𝑡) ∈ ℂ)
511508, 510eqeltrd 2825 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑡𝑅) → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) ∈ ℂ)
512 eleq1 2813 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑍 → (𝑡𝑅𝑍𝑅))
513 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑍 → ((2nd𝑝)‘𝑡) = ((2nd𝑝)‘𝑍))
514512, 513ifbieq1d 4554 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑍 → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))))
515128adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ¬ 𝑍𝑅)
516515iffalsed 4541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
5175163adant2 1128 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
51843ad2ant1 1130 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝐽 ∈ ℤ)
519518, 430zsubcld 12709 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝐽 − (1st𝑝)) ∈ ℤ)
520519zcnd 12705 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝐽 − (1st𝑝)) ∈ ℂ)
521517, 520eqeltrd 2825 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))) ∈ ℂ)
522378, 504, 505, 506, 507, 511, 514, 521fsumsplitsn 15734 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = (Σ𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) + if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝)))))
523382a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡𝑅 → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = ((2nd𝑝)‘𝑡)))
524378, 523ralrimi 3244 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ∀𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = ((2nd𝑝)‘𝑡))
525524sumeq2d 15692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = Σ𝑡𝑅 ((2nd𝑝)‘𝑡))
526 eqidd 2726 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = (2nd𝑝) → 𝑅 = 𝑅)
527 simpl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 = (2nd𝑝) ∧ 𝑡𝑅) → 𝑐 = (2nd𝑝))
528527fveq1d 6898 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 = (2nd𝑝) ∧ 𝑡𝑅) → (𝑐𝑡) = ((2nd𝑝)‘𝑡))
529526, 528sumeq12rdv 15697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = (2nd𝑝) → Σ𝑡𝑅 (𝑐𝑡) = Σ𝑡𝑅 ((2nd𝑝)‘𝑡))
530529eqeq1d 2727 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (2nd𝑝) → (Σ𝑡𝑅 (𝑐𝑡) = 𝑘 ↔ Σ𝑡𝑅 ((2nd𝑝)‘𝑡) = 𝑘))
531530elrab 3679 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡𝑅 (𝑐𝑡) = 𝑘} ↔ ((2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅) ∧ Σ𝑡𝑅 ((2nd𝑝)‘𝑡) = 𝑘))
532404, 531sylib 217 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ((2nd𝑝) ∈ ((0...𝑘) ↑m 𝑅) ∧ Σ𝑡𝑅 ((2nd𝑝)‘𝑡) = 𝑘))
533532simprd 494 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡𝑅 ((2nd𝑝)‘𝑡) = 𝑘)
534525, 533eqtrd 2765 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = 𝑘)
535507iffalsed 4541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
536535, 456eqtrd 2765 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝))) = (𝐽𝑘))
537534, 536oveq12d 7437 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (Σ𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) + if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝)))) = (𝑘 + (𝐽𝑘)))
538321sseli 3972 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℂ)
5395383ad2ant2 1131 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑘 ∈ ℂ)
540384, 298syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝐽 ∈ ℂ)
541539, 540pncan3d 11611 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑘 + (𝐽𝑘)) = 𝐽)
542537, 541eqtrd 2765 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (Σ𝑡𝑅 if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) + if(𝑍𝑅, ((2nd𝑝)‘𝑍), (𝐽 − (1st𝑝)))) = 𝐽)
543503, 522, 5423eqtrd 2769 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = 𝐽)
544493, 543jca 510 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = 𝐽))
545 eleq1w 2808 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑟 → (𝑡𝑅𝑟𝑅))
546 fveq2 6896 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑟 → ((2nd𝑝)‘𝑡) = ((2nd𝑝)‘𝑟))
547545, 546ifbieq1d 4554 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑟 → if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))) = if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))
548547cbvmptv 5262 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))
549548eqeq2i 2738 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ↔ 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))))
550549biimpi 215 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))))
551 fveq1 6895 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))) → (𝑐𝑡) = ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡))
552551sumeq2sdv 15694 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡))
553550, 552syl 17 . . . . . . . . . . . . . . 15 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡))
554553eqeq1d 2727 . . . . . . . . . . . . . 14 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = 𝐽))
555554elrab 3679 . . . . . . . . . . . . 13 ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} ↔ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))))‘𝑡) = 𝐽))
556544, 555sylibr 233 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
5575563exp 1116 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})))
558557adantr 479 . . . . . . . . . 10 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})))
559367, 368, 558rexlimd 3253 . . . . . . . . 9 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽}))
560362, 559mpd 15 . . . . . . . 8 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽})
56137eqcomd 2731 . . . . . . . . 9 (𝜑 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
562561adantr 479 . . . . . . . 8 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
563560, 562eleqtrd 2827 . . . . . . 7 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
564242a1i 11 . . . . . . . . 9 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩))
565 simpr 483 . . . . . . . . . . . . . . 15 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))))
566548a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))))
567565, 566eqtrd 2765 . . . . . . . . . . . . . 14 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝)))))
568 simpr 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟 = 𝑍) → 𝑟 = 𝑍)
569128adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟 = 𝑍) → ¬ 𝑍𝑅)
570568, 569eqneltrd 2845 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 = 𝑍) → ¬ 𝑟𝑅)
571570iffalsed 4541 . . . . . . . . . . . . . . 15 ((𝜑𝑟 = 𝑍) → if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
572571adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) ∧ 𝑟 = 𝑍) → if(𝑟𝑅, ((2nd𝑝)‘𝑟), (𝐽 − (1st𝑝))) = (𝐽 − (1st𝑝)))
57350adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
574 ovexd 7454 . . . . . . . . . . . . . 14 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝐽 − (1st𝑝)) ∈ V)
575567, 572, 573, 574fvmptd 7011 . . . . . . . . . . . . 13 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝑐𝑍) = (𝐽 − (1st𝑝)))
576575oveq2d 7435 . . . . . . . . . . . 12 ((𝜑𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝐽 − (𝑐𝑍)) = (𝐽 − (𝐽 − (1st𝑝))))
577576adantlr 713 . . . . . . . . . . 11 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝐽 − (𝑐𝑍)) = (𝐽 − (𝐽 − (1st𝑝))))
578298ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → 𝐽 ∈ ℂ)
579 nfv 1909 . . . . . . . . . . . . . . . . 17 𝑘(1st𝑝) ∈ (0...𝐽)
580 simpl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
581 simpr 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (0...𝐽) ∧ (1st𝑝) = 𝑘) → (1st𝑝) = 𝑘)
582 simpl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (0...𝐽) ∧ (1st𝑝) = 𝑘) → 𝑘 ∈ (0...𝐽))
583581, 582eqeltrd 2825 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝐽) ∧ (1st𝑝) = 𝑘) → (1st𝑝) ∈ (0...𝐽))
584580, 426, 583syl2anc 582 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (1st𝑝) ∈ (0...𝐽))
585584ex 411 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ (0...𝐽)))
586585a1i 11 . . . . . . . . . . . . . . . . 17 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ (0...𝐽))))
587366, 579, 586rexlimd 3253 . . . . . . . . . . . . . . . 16 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ (0...𝐽)))
588361, 587mpd 15 . . . . . . . . . . . . . . 15 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ (0...𝐽))
589588elfzelzd 13542 . . . . . . . . . . . . . 14 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ ℤ)
590589zcnd 12705 . . . . . . . . . . . . 13 (𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) → (1st𝑝) ∈ ℂ)
591590ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (1st𝑝) ∈ ℂ)
592578, 591nncand 11613 . . . . . . . . . . 11 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝐽 − (𝐽 − (1st𝑝))) = (1st𝑝))
593577, 592eqtrd 2765 . . . . . . . . . 10 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝐽 − (𝑐𝑍)) = (1st𝑝))
594 reseq1 5979 . . . . . . . . . . . 12 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) → (𝑐𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ↾ 𝑅))
595594adantl 480 . . . . . . . . . . 11 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝑐𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ↾ 𝑅))
59668a1i 11 . . . . . . . . . . . 12 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
597596resmptd 6045 . . . . . . . . . . 11 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ↾ 𝑅) = (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))))
598 nfv 1909 . . . . . . . . . . . . . 14 𝑘(𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝)
599382mpteq2ia 5252 . . . . . . . . . . . . . . . . . 18 (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (𝑡𝑅 ↦ ((2nd𝑝)‘𝑡))
600599a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (𝑡𝑅 ↦ ((2nd𝑝)‘𝑡)))
601409feqmptd 6966 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (2nd𝑝) = (𝑡𝑅 ↦ ((2nd𝑝)‘𝑡)))
602600, 601eqtr4d 2768 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝))
6036023exp 1116 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝))))
604603adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝))))
605367, 598, 604rexlimd 3253 . . . . . . . . . . . . 13 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝)))
606362, 605mpd 15 . . . . . . . . . . . 12 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝))
607606adantr 479 . . . . . . . . . . 11 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝑡𝑅 ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) = (2nd𝑝))
608595, 597, 6073eqtrd 2769 . . . . . . . . . 10 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → (𝑐𝑅) = (2nd𝑝))
609593, 608opeq12d 4883 . . . . . . . . 9 (((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) → ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
610 opex 5466 . . . . . . . . . 10 ⟨(1st𝑝), (2nd𝑝)⟩ ∈ V
611610a1i 11 . . . . . . . . 9 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → ⟨(1st𝑝), (2nd𝑝)⟩ ∈ V)
612564, 609, 563, 611fvmptd 7011 . . . . . . . 8 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))) = ⟨(1st𝑝), (2nd𝑝)⟩)
613 nfv 1909 . . . . . . . . . 10 𝑘⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝
614 1st2nd2 8033 . . . . . . . . . . . . 13 (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
615614eqcomd 2731 . . . . . . . . . . . 12 (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → ⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝)
616615a1i 11 . . . . . . . . . . 11 (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → ⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝))
617616a1i 11 . . . . . . . . . 10 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → ⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝)))
618367, 613, 617rexlimd 3253 . . . . . . . . 9 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶𝑅)‘𝑘)) → ⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝))
619362, 618mpd 15 . . . . . . . 8 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → ⟨(1st𝑝), (2nd𝑝)⟩ = 𝑝)
620612, 619eqtr2d 2766 . . . . . . 7 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))))
621 fveq2 6896 . . . . . . . 8 (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) → (𝐷𝑐) = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝))))))
622621rspceeqv 3628 . . . . . . 7 (((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡𝑅, ((2nd𝑝)‘𝑡), (𝐽 − (1st𝑝)))))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷𝑐))
623563, 620, 622syl2anc 582 . . . . . 6 ((𝜑𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷𝑐))
624623ralrimiva 3135 . . . . 5 (𝜑 → ∀𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷𝑐))
625243, 624jca 510 . . . 4 (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ ∀𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷𝑐)))
626 dffo3 7111 . . . 4 (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ ∀𝑝 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷𝑐)))
627625, 626sylibr 233 . . 3 (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
628359, 627jca 510 . 2 (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))))
629 df-f1o 6556 . 2 (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘))))
630628, 629sylibr 233 1 (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cun 3942  wss 3944  ifcif 4530  𝒫 cpw 4604  {csn 4630  cop 4636   ciun 4997   class class class wbr 5149  cmpt 5232   × cxp 5676  cres 5680   Fn wfn 6544  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  m cmap 8845  Fincfn 8964  cc 11143  cr 11144  0cc0 11145   + caddc 11148  cle 11286  cmin 11481  0cn0 12510  cz 12596  ...cfz 13524  Σcsu 15676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-oi 9540  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-ico 13370  df-fz 13525  df-fzo 13668  df-seq 14008  df-exp 14068  df-hash 14334  df-cj 15090  df-re 15091  df-im 15092  df-sqrt 15226  df-abs 15227  df-clim 15476  df-sum 15677
This theorem is referenced by:  dvnprodlem2  45478
  Copyright terms: Public domain W3C validator