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Theorem psrass1lem 19317
Description: A group sum commutation used by psrass1 19345. (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 19315 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}))
8 gsumbagdiag.x . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑋𝐵)
98anassrs 679 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑋𝐵)
10 eqid 2621 . . . . . . . . . . 11 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
119, 10fmptd 6351 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
123adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → 𝐼𝑉)
13 ssrab2 3672 . . . . . . . . . . . . . 14 {𝑦𝐷𝑦𝑟𝐹} ⊆ 𝐷
142, 13eqsstri 3620 . . . . . . . . . . . . 13 𝑆𝐷
154adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝐹𝐷)
16 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝑗𝑆)
171, 2psrbagconcl 19313 . . . . . . . . . . . . . 14 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1812, 15, 16, 17syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1914, 18sseldi 3586 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝐷)
20 eqid 2621 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
211, 20psrbagconf1o 19314 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2212, 19, 21syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
23 f1of 6104 . . . . . . . . . . 11 ((𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2422, 23syl 17 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
25 fco 6025 . . . . . . . . . 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)
2611, 24, 25syl2anc 692 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
2712adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐼𝑉)
2815adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹𝐷)
291psrbagf 19305 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
3027, 28, 29syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹:𝐼⟶ℕ0)
3130ffvelrnda 6325 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3216adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝑆)
3314, 32sseldi 3586 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝐷)
341psrbagf 19305 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3527, 33, 34syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗:𝐼⟶ℕ0)
3635ffvelrnda 6325 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
37 ssrab2 3672 . . . . . . . . . . . . . . . . . 18 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ⊆ 𝐷
38 simpr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
3937, 38sseldi 3586 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚𝐷)
401psrbagf 19305 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
4127, 39, 40syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚:𝐼⟶ℕ0)
4241ffvelrnda 6325 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
43 nn0cn 11262 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
44 nn0cn 11262 . . . . . . . . . . . . . . . 16 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
45 nn0cn 11262 . . . . . . . . . . . . . . . 16 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
46 sub32 10275 . . . . . . . . . . . . . . . 16 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4743, 44, 45, 46syl3an 1365 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4831, 36, 42, 47syl3anc 1323 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4948mpteq2dva 4714 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
50 ovexd 6645 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
5130feqmptd 6216 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5235feqmptd 6216 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5327, 31, 36, 51, 52offval2 6879 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5441feqmptd 6216 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5527, 50, 42, 53, 54offval2 6879 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
56 ovexd 6645 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5727, 31, 42, 51, 54offval2 6879 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5827, 56, 36, 57, 52offval2 6879 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5949, 55, 583eqtr4d 2665 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
6019adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) ∈ 𝐷)
611, 20psrbagconcl 19313 . . . . . . . . . . . . 13 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6227, 60, 38, 61syl3anc 1323 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6359, 62eqeltrrd 2699 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6459mpteq2dva 4714 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗)))
65 nfcv 2761 . . . . . . . . . . . . 13 𝑛𝑋
66 nfcsb1v 3535 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑋
67 csbeq1a 3528 . . . . . . . . . . . . 13 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6865, 66, 67cbvmpt 4719 . . . . . . . . . . . 12 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋)
6968a1i 11 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋))
70 csbeq1 3522 . . . . . . . . . . 11 (𝑛 = ((𝐹𝑓𝑚) ∘𝑓𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7163, 64, 69, 70fmptco 6362 . . . . . . . . . 10 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
7271feq1d 5997 . . . . . . . . 9 ((𝜑𝑗𝑆) → (((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵))
7326, 72mpbid 222 . . . . . . . 8 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
74 eqid 2621 . . . . . . . . 9 (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7574fmpt 6347 . . . . . . . 8 (∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
7673, 75sylibr 224 . . . . . . 7 ((𝜑𝑗𝑆) → ∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7776r19.21bi 2928 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7877anasss 678 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
797, 78syldan 487 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
801, 2, 3, 4, 5, 6, 79gsumbagdiag 19316 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
81 eqid 2621 . . . 4 (0g𝐺) = (0g𝐺)
821psrbaglefi 19312 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
833, 4, 82syl2anc 692 . . . . 5 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
842, 83syl5eqel 2702 . . . 4 (𝜑𝑆 ∈ Fin)
853adantr 481 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
864adantr 481 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
87 simpr 477 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
881, 2psrbagconcl 19313 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
8985, 86, 87, 88syl3anc 1323 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
9014, 89sseldi 3586 . . . . 5 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝐷)
911psrbaglefi 19312 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑚) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
9285, 90, 91syl2anc 692 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
93 xpfi 8191 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
9484, 84, 93syl2anc 692 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
95 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚𝑆)
967simpld 475 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑗𝑆)
97 brxp 5117 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
9895, 96, 97sylanbrc 697 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
9998pm2.24d 147 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
10099impr 648 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1015, 81, 6, 84, 92, 79, 94, 100gsum2d2 18313 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1021psrbaglefi 19312 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
10312, 19, 102syl2anc 692 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
104 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗𝑆)
1051, 2, 3, 4gsumbagdiaglem 19315 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}))
106105simpld 475 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑚𝑆)
107 brxp 5117 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
108104, 106, 107sylanbrc 697 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
109108pm2.24d 147 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
110109impr 648 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1115, 81, 6, 84, 103, 78, 94, 110gsum2d2 18313 . . 3 (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
11280, 101, 1113eqtr3d 2663 . 2 (𝜑 → (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1136adantr 481 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
11479anassrs 679 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
115 eqid 2621 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
116114, 115fmptd 6351 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}⟶𝐵)
117 ovex 6643 . . . . . . . . . . . 12 (ℕ0𝑚 𝐼) ∈ V
1181, 117rabex2 4785 . . . . . . . . . . 11 𝐷 ∈ V
119118a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
120 rabexg 4782 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V)
121 mptexg 6449 . . . . . . . . . 10 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
122119, 120, 1213syl 18 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
123 funmpt 5894 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
125 fvexd 6170 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
126 suppssdm 7268 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
127115dmmptss 5600 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
128126, 127sstri 3597 . . . . . . . . . 10 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
129128a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
130 suppssfifsupp 8250 . . . . . . . . 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))
131122, 124, 125, 92, 129, 130syl32anc 1331 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1325, 81, 113, 92, 116, 131gsumcl 18256 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) ∈ 𝐵)
133 eqid 2621 . . . . . . 7 (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
134132, 133fmptd 6351 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))):𝑆𝐵)
1351, 2psrbagconf1o 19314 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
1363, 4, 135syl2anc 692 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
137 f1ocnv 6116 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
138 f1of 6104 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
139136, 137, 1383syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
140 fco 6025 . . . . . 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)
141134, 139, 140syl2anc 692 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵)
142 coass 5623 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
143 f1ococnv2 6130 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
144136, 143syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
145144coeq2d 5254 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
146142, 145syl5eq 2667 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
147 eqidd 2622 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)) = (𝑚𝑆 ↦ (𝐹𝑓𝑚)))
148 eqidd 2622 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
149 breq2 4627 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑥𝑟𝑛𝑥𝑟 ≤ (𝐹𝑓𝑚)))
150149rabbidv 3181 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → {𝑥𝐷𝑥𝑟𝑛} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
151 ovex 6643 . . . . . . . . . . . . 13 (𝑛𝑓𝑗) ∈ V
152 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛𝑓𝑗) → 𝑋 = 𝑌)
153151, 152csbie 3545 . . . . . . . . . . . 12 (𝑛𝑓𝑗) / 𝑘𝑋 = 𝑌
154 oveq1 6622 . . . . . . . . . . . . 13 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
155154csbeq1d 3526 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
156153, 155syl5eqr 2669 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → 𝑌 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
157150, 156mpteq12dv 4703 . . . . . . . . . 10 (𝑛 = (𝐹𝑓𝑚) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
158157oveq2d 6631 . . . . . . . . 9 (𝑛 = (𝐹𝑓𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
15989, 147, 148, 158fmptco 6362 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
160159coeq1d 5253 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
161 coires1 5622 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆)
162 ssid 3609 . . . . . . . . . 10 𝑆𝑆
163 resmpt 5418 . . . . . . . . . 10 (𝑆𝑆 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
164162, 163ax-mp 5 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
165161, 164eqtri 2643 . . . . . . . 8 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
166165a1i 11 . . . . . . 7 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
167146, 160, 1663eqtr3d 2663 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
168167feq1d 5997 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵))
169141, 168mpbid 222 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵)
170 rabexg 4782 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
171118, 170mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
1722, 171syl5eqel 2702 . . . . . 6 (𝜑𝑆 ∈ V)
173 mptexg 6449 . . . . . 6 (𝑆 ∈ V → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
174172, 173syl 17 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
175 funmpt 5894 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
176175a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
177 fvexd 6170 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
178 suppssdm 7268 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
179 eqid 2621 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
180179dmmptss 5600 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ⊆ 𝑆
181178, 180sstri 3597 . . . . . 6 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆
182181a1i 11 . . . . 5 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)
183 suppssfifsupp 8250 . . . . 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))
184174, 176, 177, 84, 182, 183syl32anc 1331 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1855, 81, 6, 84, 169, 184, 136gsumf1o 18257 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))))
186159oveq2d 6631 . . 3 (𝜑 → (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
187185, 186eqtrd 2655 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1886adantr 481 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
189118a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
190 rabexg 4782 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V)
191 mptexg 6449 . . . . . . . 8 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
192189, 190, 1913syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
193 funmpt 5894 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
194193a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))
195 fvexd 6170 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
196 suppssdm 7268 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
19710dmmptss 5600 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
198196, 197sstri 3597 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
199198a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
200 suppssfifsupp 8250 . . . . . . 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))
201192, 194, 195, 103, 199, 200syl32anc 1331 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) finSupp (0g𝐺))
2025, 81, 188, 103, 11, 201, 22gsumf1o 18257 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))))
20371oveq2d 6631 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
204202, 203eqtrd 2655 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
205204mpteq2dva 4714 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
206205oveq2d 6631 . 2 (𝜑 → (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
207112, 187, 2063eqtr4d 2665 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 1480  wcel 1987  wral 2908  {crab 2912  Vcvv 3190  csb 3519  wss 3560   class class class wbr 4623  cmpt 4683   I cid 4994   × cxp 5082  ccnv 5083  dom cdm 5084  cres 5086  cima 5087  ccom 5088  Fun wfun 5851  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cmpt2 6617  𝑓 cof 6860  𝑟 cofr 6861   supp csupp 7255  𝑚 cmap 7817  Fincfn 7915   finSupp cfsupp 8235  cc 9894  cle 10035  cmin 10226  cn 10980  0cn0 11252  Basecbs 15800  0gc0g 16040   Σg cgsu 16041  CMndccmn 18133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-gsum 16043  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-mulg 17481  df-cntz 17690  df-cmn 18135
This theorem is referenced by:  psrass1  19345
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