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Theorem psrass1lem 21345
Description: A group sum commutation used by psrass1 21374. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
Hypotheses
Ref Expression
gsumbagdiag.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
gsumbagdiag.s 𝑆 = {𝑦𝐷𝑦r𝐹}
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:   𝑥,𝐷   𝑦,𝐷   𝑓,𝐹,𝑥   𝑦,𝐹   𝑓,𝐼   𝑓,𝑋,𝑥   𝑦,𝑋   𝑓,𝑌,𝑥   𝑦,𝑌   𝐵,𝑗,𝑘   𝐷,𝑗,𝑘   𝑗,𝐹,𝑘   𝑗,𝐺,𝑘   𝑦,𝐼,𝑓   𝑆,𝑗,𝑘   𝜑,𝑗,𝑘   𝑓,𝑗,𝑘,𝑦   𝑥,𝑗,𝑘   𝐷,𝑛,𝑗,𝑘,𝑥   𝑥,𝑓   𝑛,𝐹   𝑛,𝐺   𝑥,𝐼   𝑆,𝑛   𝑛,𝑋   𝑘,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑛)   𝐷(𝑓)   𝑆(𝑥,𝑦,𝑓)   𝐺(𝑥,𝑦,𝑓)   𝐼(𝑗,𝑘,𝑛)   𝑋(𝑗,𝑘)   𝑌(𝑗,𝑛)

Proof of Theorem psrass1lem
Dummy variables 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumbagdiag.d . . . 4 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
2 gsumbagdiag.s . . . 4 𝑆 = {𝑦𝐷𝑦r𝐹}
3 gsumbagdiag.f . . . 4 (𝜑𝐹𝐷)
4 gsumbagdiag.b . . . 4 𝐵 = (Base‘𝐺)
5 gsumbagdiag.g . . . 4 (𝜑𝐺 ∈ CMnd)
61, 2, 3gsumbagdiaglem 21343 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}))
7 gsumbagdiag.x . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)
87anassrs 468 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑋𝐵)
98fmpttd 7063 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
102ssrab3 4040 . . . . . . . . . . . 12 𝑆𝐷
111, 2psrbagconcl 21336 . . . . . . . . . . . . 13 ((𝐹𝐷𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
123, 11sylan 580 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1310, 12sselid 3942 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝐷)
14 eqid 2736 . . . . . . . . . . . 12 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} = {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
151, 14psrbagconf1o 21338 . . . . . . . . . . 11 ((𝐹f𝑗) ∈ 𝐷 → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
1613, 15syl 17 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
17 f1of 6784 . . . . . . . . . 10 ((𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)} → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
1816, 17syl 17 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
199, 18fcod 6694 . . . . . . . 8 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
203adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑆) → 𝐹𝐷)
2120adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹𝐷)
221psrbagf 21320 . . . . . . . . . . . . . . . 16 (𝐹𝐷𝐹:𝐼⟶ℕ0)
2321, 22syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹:𝐼⟶ℕ0)
2423ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
25 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝑆)
2610, 25sselid 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝐷)
271psrbagf 21320 . . . . . . . . . . . . . . . 16 (𝑗𝐷𝑗:𝐼⟶ℕ0)
2826, 27syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗:𝐼⟶ℕ0)
2928ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
30 ssrab2 4037 . . . . . . . . . . . . . . . . 17 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ⊆ 𝐷
31 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
3230, 31sselid 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚𝐷)
331psrbagf 21320 . . . . . . . . . . . . . . . 16 (𝑚𝐷𝑚:𝐼⟶ℕ0)
3432, 33syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚:𝐼⟶ℕ0)
3534ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
36 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
37 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
38 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
39 sub32 11435 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4036, 37, 38, 39syl3an 1160 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4124, 29, 35, 40syl3anc 1371 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4241mpteq2dva 5205 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
4334ffnd 6669 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 Fn 𝐼)
4431, 43fndmexd 7843 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐼 ∈ V)
45 ovexd 7392 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
4623feqmptd 6910 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
4728feqmptd 6910 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
4844, 24, 29, 46, 47offval2 7637 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
4934feqmptd 6910 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5044, 45, 35, 48, 49offval2 7637 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
51 ovexd 7392 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5244, 24, 35, 46, 49offval2 7637 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5344, 51, 29, 52, 47offval2 7637 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5442, 50, 533eqtr4d 2786 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = ((𝐹f𝑚) ∘f𝑗))
551, 14psrbagconcl 21336 . . . . . . . . . . . 12 (((𝐹f𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5613, 55sylan 580 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5754, 56eqeltrrd 2839 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5854mpteq2dva 5205 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗)))
59 nfcv 2907 . . . . . . . . . . . 12 𝑛𝑋
60 nfcsb1v 3880 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝑋
61 csbeq1a 3869 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6259, 60, 61cbvmpt 5216 . . . . . . . . . . 11 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑛 / 𝑘𝑋)
6362a1i 11 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑛 / 𝑘𝑋))
64 csbeq1 3858 . . . . . . . . . 10 (𝑛 = ((𝐹f𝑚) ∘f𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
6557, 58, 63, 64fmptco 7075 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
6665feq1d 6653 . . . . . . . 8 ((𝜑𝑗𝑆) → (((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵))
6719, 66mpbid 231 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
6867fvmptelcdm 7061 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
6968anasss 467 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
706, 69syldan 591 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
711, 2, 3, 4, 5, 70gsumbagdiag 21344 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
72 eqid 2736 . . . 4 (0g𝐺) = (0g𝐺)
731psrbaglefi 21334 . . . . . 6 (𝐹𝐷 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
743, 73syl 17 . . . . 5 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
752, 74eqeltrid 2842 . . . 4 (𝜑𝑆 ∈ Fin)
761, 2psrbagconcl 21336 . . . . . . 7 ((𝐹𝐷𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
773, 76sylan 580 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
7810, 77sselid 3942 . . . . 5 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝐷)
791psrbaglefi 21334 . . . . 5 ((𝐹f𝑚) ∈ 𝐷 → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
8078, 79syl 17 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
81 xpfi 9261 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
8275, 75, 81syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
83 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚𝑆)
846simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑗𝑆)
85 brxp 5681 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
8683, 84, 85sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
8786pm2.24d 151 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
8887impr 455 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
894, 72, 5, 75, 80, 70, 82, 88gsum2d2 19751 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
901psrbaglefi 21334 . . . . 5 ((𝐹f𝑗) ∈ 𝐷 → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
9113, 90syl 17 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
92 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗𝑆)
931, 2, 3gsumbagdiaglem 21343 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}))
9493simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑚𝑆)
95 brxp 5681 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
9692, 94, 95sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
9796pm2.24d 151 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
9897impr 455 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
994, 72, 5, 75, 91, 69, 82, 98gsum2d2 19751 . . 3 (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
10071, 89, 993eqtr3d 2784 . 2 (𝜑 → (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
1015adantr 481 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
10270anassrs 468 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
103102fmpttd 7063 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑚)}⟶𝐵)
104 ovex 7390 . . . . . . . . . . . 12 (ℕ0m 𝐼) ∈ V
1051, 104rabex2 5291 . . . . . . . . . . 11 𝐷 ∈ V
106105a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
107 rabexg 5288 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V)
108 mptexg 7171 . . . . . . . . . 10 ({𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ∈ V)
109106, 107, 1083syl 18 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ∈ V)
110 funmpt 6539 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
111110a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
112 fvexd 6857 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
113 suppssdm 8108 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
114 eqid 2736 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
115114dmmptss 6193 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
116113, 115sstri 3953 . . . . . . . . . 10 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
117116a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
118 suppssfifsupp 9320 . . . . . . . . 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))
119109, 111, 112, 80, 117, 118syl32anc 1378 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1204, 72, 101, 80, 103, 119gsumcl 19692 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) ∈ 𝐵)
121120fmpttd 7063 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))):𝑆𝐵)
1221, 2psrbagconf1o 21338 . . . . . . . 8 (𝐹𝐷 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
1233, 122syl 17 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
124 f1ocnv 6796 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
125 f1of 6784 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
126123, 124, 1253syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
127121, 126fcod 6694 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵)
128 coass 6217 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
129 f1ococnv2 6811 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
130123, 129syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
131130coeq2d 5818 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
132128, 131eqtrid 2788 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
133 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)) = (𝑚𝑆 ↦ (𝐹f𝑚)))
134 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
135 breq2 5109 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑥r𝑛𝑥r ≤ (𝐹f𝑚)))
136135rabbidv 3415 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → {𝑥𝐷𝑥r𝑛} = {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
137 ovex 7390 . . . . . . . . . . . . 13 (𝑛f𝑗) ∈ V
138 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)
139137, 138csbie 3891 . . . . . . . . . . . 12 (𝑛f𝑗) / 𝑘𝑋 = 𝑌
140 oveq1 7364 . . . . . . . . . . . . 13 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) = ((𝐹f𝑚) ∘f𝑗))
141140csbeq1d 3859 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
142139, 141eqtr3id 2790 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → 𝑌 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
143136, 142mpteq12dv 5196 . . . . . . . . . 10 (𝑛 = (𝐹f𝑚) → (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
144143oveq2d 7373 . . . . . . . . 9 (𝑛 = (𝐹f𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
14577, 133, 134, 144fmptco 7075 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
146145coeq1d 5817 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
147 coires1 6216 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆)
148 ssid 3966 . . . . . . . . . 10 𝑆𝑆
149 resmpt 5991 . . . . . . . . . 10 (𝑆𝑆 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
150148, 149ax-mp 5 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
151147, 150eqtri 2764 . . . . . . . 8 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
152151a1i 11 . . . . . . 7 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
153132, 146, 1523eqtr3d 2784 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
154153feq1d 6653 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵))
155127, 154mpbid 231 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵)
156 rabexg 5288 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦r𝐹} ∈ V)
157105, 156mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ V)
1582, 157eqeltrid 2842 . . . . . 6 (𝜑𝑆 ∈ V)
159158mptexd 7174 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∈ V)
160 funmpt 6539 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
161160a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
162 fvexd 6857 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
163 suppssdm 8108 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
164 eqid 2736 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
165164dmmptss 6193 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ⊆ 𝑆
166163, 165sstri 3953 . . . . . 6 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆
167166a1i 11 . . . . 5 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)
168 suppssfifsupp 9320 . . . . 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))
169159, 161, 162, 75, 167, 168syl32anc 1378 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1704, 72, 5, 75, 155, 169, 123gsumf1o 19693 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))))
171145oveq2d 7373 . . 3 (𝜑 → (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
172170, 171eqtrd 2776 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
1735adantr 481 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
174105a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
175 rabexg 5288 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V)
176 mptexg 7171 . . . . . . . 8 ({𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
177174, 175, 1763syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
178 funmpt 6539 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
179178a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))
180 fvexd 6857 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
181 suppssdm 8108 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
182 eqid 2736 . . . . . . . . . 10 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
183182dmmptss 6193 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
184181, 183sstri 3953 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
185184a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
186 suppssfifsupp 9320 . . . . . . 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))
187177, 179, 180, 91, 185, 186syl32anc 1378 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) finSupp (0g𝐺))
1884, 72, 173, 91, 9, 187, 16gsumf1o 19693 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))))
18965oveq2d 7373 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
190188, 189eqtrd 2776 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
191190mpteq2dva 5205 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
192191oveq2d 7373 . 2 (𝜑 → (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
193100, 172, 1923eqtr4d 2786 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 396   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  csb 3855  wss 3910   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  dom cdm 5633  cres 5635  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  r cofr 7616   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  cc 11049  cle 11190  cmin 11385  cn 12153  0cn0 12413  Basecbs 17083  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564
This theorem is referenced by:  psrass1  21374
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