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Theorem psrass1lem 19651
Description: A group sum commutation used by psrass1 19679. (Contributed by Mario Carneiro, 5-Jan-2015.)
Hypotheses
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.1 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
gsumbagdiag.i (𝜑𝐼𝑉)
gsumbagdiag.f (𝜑𝐹𝐷)
gsumbagdiag.b 𝐵 = (Base‘𝐺)
gsumbagdiag.g (𝜑𝐺 ∈ CMnd)
gsumbagdiag.x ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑋𝐵)
psrass1lem.y (𝑘 = (𝑛𝑓𝑗) → 𝑋 = 𝑌)
Assertion
Ref Expression
psrass1lem (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)))))
Distinct variable groups:   𝑓,𝑗,𝑘,𝑛,𝑥,𝑦,𝐹   𝑓,𝐺,𝑗,𝑘,𝑛,𝑥,𝑦   𝑛,𝑉,𝑥,𝑦   𝑓,𝐼,𝑛,𝑥,𝑦   𝜑,𝑗,𝑘   𝑆,𝑗,𝑘,𝑛,𝑥   𝐵,𝑗,𝑘   𝐷,𝑗,𝑘,𝑛,𝑥,𝑦   𝑓,𝑋,𝑛,𝑥,𝑦   𝑓,𝑌,𝑘,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑛)   𝐷(𝑓)   𝑆(𝑦,𝑓)   𝐼(𝑗,𝑘)   𝑉(𝑓,𝑗,𝑘)   𝑋(𝑗,𝑘)   𝑌(𝑗,𝑛)

Proof of Theorem psrass1lem
Dummy variables 𝑚 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrbag.d . . . 4 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
2 psrbagconf1o.1 . . . 4 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
3 gsumbagdiag.i . . . 4 (𝜑𝐼𝑉)
4 gsumbagdiag.f . . . 4 (𝜑𝐹𝐷)
5 gsumbagdiag.b . . . 4 𝐵 = (Base‘𝐺)
6 gsumbagdiag.g . . . 4 (𝜑𝐺 ∈ CMnd)
71, 2, 3, 4gsumbagdiaglem 19649 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}))
8 gsumbagdiag.x . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑋𝐵)
98anassrs 459 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑋𝐵)
109fmpttd 6575 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
113adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → 𝐼𝑉)
12 ssrab2 3847 . . . . . . . . . . . . . 14 {𝑦𝐷𝑦𝑟𝐹} ⊆ 𝐷
132, 12eqsstri 3795 . . . . . . . . . . . . 13 𝑆𝐷
144adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝐹𝐷)
15 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝑗𝑆)
161, 2psrbagconcl 19647 . . . . . . . . . . . . . 14 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1711, 14, 15, 16syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1813, 17sseldi 3759 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝐷)
19 eqid 2765 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
201, 19psrbagconf1o 19648 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2111, 18, 20syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
22 f1of 6320 . . . . . . . . . . 11 ((𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2321, 22syl 17 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
24 fco 6240 . . . . . . . . . 10 (((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵 ∧ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
2510, 23, 24syl2anc 579 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
2611adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐼𝑉)
2714adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹𝐷)
281psrbagf 19639 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
2926, 27, 28syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹:𝐼⟶ℕ0)
3029ffvelrnda 6549 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3115adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝑆)
3213, 31sseldi 3759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝐷)
331psrbagf 19639 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3426, 32, 33syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗:𝐼⟶ℕ0)
3534ffvelrnda 6549 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
36 ssrab2 3847 . . . . . . . . . . . . . . . . . 18 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ⊆ 𝐷
37 simpr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
3836, 37sseldi 3759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚𝐷)
391psrbagf 19639 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
4026, 38, 39syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚:𝐼⟶ℕ0)
4140ffvelrnda 6549 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
42 nn0cn 11549 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
43 nn0cn 11549 . . . . . . . . . . . . . . . 16 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
44 nn0cn 11549 . . . . . . . . . . . . . . . 16 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
45 sub32 10569 . . . . . . . . . . . . . . . 16 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4642, 43, 44, 45syl3an 1199 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4730, 35, 41, 46syl3anc 1490 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4847mpteq2dva 4903 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
49 ovexd 6876 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
5029feqmptd 6438 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5134feqmptd 6438 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5226, 30, 35, 50, 51offval2 7112 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5340feqmptd 6438 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5426, 49, 41, 52, 53offval2 7112 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
55 ovexd 6876 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5626, 30, 41, 50, 53offval2 7112 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5726, 55, 35, 56, 51offval2 7112 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5848, 54, 573eqtr4d 2809 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
5918adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) ∈ 𝐷)
601, 19psrbagconcl 19647 . . . . . . . . . . . . 13 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6126, 59, 37, 60syl3anc 1490 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6258, 61eqeltrrd 2845 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6358mpteq2dva 4903 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗)))
64 nfcv 2907 . . . . . . . . . . . . 13 𝑛𝑋
65 nfcsb1v 3707 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑋
66 csbeq1a 3700 . . . . . . . . . . . . 13 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6764, 65, 66cbvmpt 4908 . . . . . . . . . . . 12 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋)
6867a1i 11 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋))
69 csbeq1 3694 . . . . . . . . . . 11 (𝑛 = ((𝐹𝑓𝑚) ∘𝑓𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7062, 63, 68, 69fmptco 6587 . . . . . . . . . 10 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
7170feq1d 6208 . . . . . . . . 9 ((𝜑𝑗𝑆) → (((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵))
7225, 71mpbid 223 . . . . . . . 8 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
73 eqid 2765 . . . . . . . . 9 (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7473fmpt 6570 . . . . . . . 8 (∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
7572, 74sylibr 225 . . . . . . 7 ((𝜑𝑗𝑆) → ∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7675r19.21bi 3079 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7776anasss 458 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
787, 77syldan 585 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
791, 2, 3, 4, 5, 6, 78gsumbagdiag 19650 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
80 eqid 2765 . . . 4 (0g𝐺) = (0g𝐺)
811psrbaglefi 19646 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
823, 4, 81syl2anc 579 . . . . 5 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
832, 82syl5eqel 2848 . . . 4 (𝜑𝑆 ∈ Fin)
843adantr 472 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
854adantr 472 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
86 simpr 477 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
871, 2psrbagconcl 19647 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
8884, 85, 86, 87syl3anc 1490 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
8913, 88sseldi 3759 . . . . 5 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝐷)
901psrbaglefi 19646 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑚) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
9184, 89, 90syl2anc 579 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
92 xpfi 8438 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
9383, 83, 92syl2anc 579 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
94 simprl 787 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚𝑆)
957simpld 488 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑗𝑆)
96 brxp 5323 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
9794, 95, 96sylanbrc 578 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
9897pm2.24d 148 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
9998impr 446 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1005, 80, 6, 83, 91, 78, 93, 99gsum2d2 18639 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1011psrbaglefi 19646 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
10211, 18, 101syl2anc 579 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
103 simprl 787 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗𝑆)
1041, 2, 3, 4gsumbagdiaglem 19649 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}))
105104simpld 488 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑚𝑆)
106 brxp 5323 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
107103, 105, 106sylanbrc 578 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
108107pm2.24d 148 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
109108impr 446 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1105, 80, 6, 83, 102, 77, 93, 109gsum2d2 18639 . . 3 (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
11179, 100, 1103eqtr3d 2807 . 2 (𝜑 → (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1126adantr 472 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
11378anassrs 459 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
114113fmpttd 6575 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}⟶𝐵)
115 ovex 6874 . . . . . . . . . . . 12 (ℕ0𝑚 𝐼) ∈ V
1161, 115rabex2 4975 . . . . . . . . . . 11 𝐷 ∈ V
117116a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
118 rabexg 4972 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V)
119 mptexg 6677 . . . . . . . . . 10 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
120117, 118, 1193syl 18 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
121 funmpt 6106 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
122121a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
123 fvexd 6390 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
124 suppssdm 7510 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
125 eqid 2765 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
126125dmmptss 5817 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
127124, 126sstri 3770 . . . . . . . . . 10 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
128127a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
129 suppssfifsupp 8497 . . . . . . . . 9 ((((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V ∧ Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∧ (0g𝐺) ∈ V) ∧ ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin ∧ ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) finSupp (0g𝐺))
130120, 122, 123, 91, 128, 129syl32anc 1497 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1315, 80, 112, 91, 114, 130gsumcl 18582 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) ∈ 𝐵)
132131fmpttd 6575 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))):𝑆𝐵)
1331, 2psrbagconf1o 19648 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
1343, 4, 133syl2anc 579 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
135 f1ocnv 6332 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
136 f1of 6320 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
137134, 135, 1363syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
138 fco 6240 . . . . . 6 (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))):𝑆𝐵(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆) → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵)
139132, 137, 138syl2anc 579 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵)
140 coass 5840 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
141 f1ococnv2 6346 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
142134, 141syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
143142coeq2d 5453 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
144140, 143syl5eq 2811 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
145 eqidd 2766 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)) = (𝑚𝑆 ↦ (𝐹𝑓𝑚)))
146 eqidd 2766 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
147 breq2 4813 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑥𝑟𝑛𝑥𝑟 ≤ (𝐹𝑓𝑚)))
148147rabbidv 3338 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → {𝑥𝐷𝑥𝑟𝑛} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
149 ovex 6874 . . . . . . . . . . . . 13 (𝑛𝑓𝑗) ∈ V
150 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛𝑓𝑗) → 𝑋 = 𝑌)
151149, 150csbie 3717 . . . . . . . . . . . 12 (𝑛𝑓𝑗) / 𝑘𝑋 = 𝑌
152 oveq1 6849 . . . . . . . . . . . . 13 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
153152csbeq1d 3698 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
154151, 153syl5eqr 2813 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → 𝑌 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
155148, 154mpteq12dv 4892 . . . . . . . . . 10 (𝑛 = (𝐹𝑓𝑚) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
156155oveq2d 6858 . . . . . . . . 9 (𝑛 = (𝐹𝑓𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
15788, 145, 146, 156fmptco 6587 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
158157coeq1d 5452 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
159 coires1 5839 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆)
160 ssid 3783 . . . . . . . . . 10 𝑆𝑆
161 resmpt 5626 . . . . . . . . . 10 (𝑆𝑆 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
162160, 161ax-mp 5 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
163159, 162eqtri 2787 . . . . . . . 8 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
164163a1i 11 . . . . . . 7 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
165144, 158, 1643eqtr3d 2807 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
166165feq1d 6208 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵))
167139, 166mpbid 223 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵)
168 rabexg 4972 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
169116, 168mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
1702, 169syl5eqel 2848 . . . . . 6 (𝜑𝑆 ∈ V)
171 mptexg 6677 . . . . . 6 (𝑆 ∈ V → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
172170, 171syl 17 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
173 funmpt 6106 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
174173a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
175 fvexd 6390 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
176 suppssdm 7510 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
177 eqid 2765 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
178177dmmptss 5817 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ⊆ 𝑆
179176, 178sstri 3770 . . . . . 6 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆
180179a1i 11 . . . . 5 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)
181 suppssfifsupp 8497 . . . . 5 ((((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V ∧ Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∧ (0g𝐺) ∈ V) ∧ (𝑆 ∈ Fin ∧ ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)) → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) finSupp (0g𝐺))
182172, 174, 175, 83, 180, 181syl32anc 1497 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1835, 80, 6, 83, 167, 182, 134gsumf1o 18583 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))))
184157oveq2d 6858 . . 3 (𝜑 → (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
185183, 184eqtrd 2799 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1866adantr 472 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
187116a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
188 rabexg 4972 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V)
189 mptexg 6677 . . . . . . . 8 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
190187, 188, 1893syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
191 funmpt 6106 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
192191a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))
193 fvexd 6390 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
194 suppssdm 7510 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
195 eqid 2765 . . . . . . . . . 10 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
196195dmmptss 5817 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
197194, 196sstri 3770 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
198197a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
199 suppssfifsupp 8497 . . . . . . 7 ((((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V ∧ Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∧ (0g𝐺) ∈ V) ∧ ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin ∧ ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) finSupp (0g𝐺))
200190, 192, 193, 102, 198, 199syl32anc 1497 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) finSupp (0g𝐺))
2015, 80, 186, 102, 10, 200, 21gsumf1o 18583 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))))
20270oveq2d 6858 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
203201, 202eqtrd 2799 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
204203mpteq2dva 4903 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
205204oveq2d 6858 . 2 (𝜑 → (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
206111, 185, 2053eqtr4d 2809 1 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  csb 3691  wss 3732   class class class wbr 4809  cmpt 4888   I cid 5184   × cxp 5275  ccnv 5276  dom cdm 5277  cres 5279  cima 5280  ccom 5281  Fun wfun 6062  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑓 cof 7093  𝑟 cofr 7094   supp csupp 7497  𝑚 cmap 8060  Fincfn 8160   finSupp cfsupp 8482  cc 10187  cle 10329  cmin 10520  cn 11274  0cn0 11538  Basecbs 16130  0gc0g 16366   Σg cgsu 16367  CMndccmn 18459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  df-cmn 18461
This theorem is referenced by:  psrass1  19679
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