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Theorem psrass1lem 20756
Description: A group sum commutation used by psrass1 20784. (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 20754 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}))
7 gsumbagdiag.x . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)
87anassrs 471 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑋𝐵)
98fmpttd 6889 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
102ssrab3 3971 . . . . . . . . . . . 12 𝑆𝐷
111, 2psrbagconcl 20747 . . . . . . . . . . . . 13 ((𝐹𝐷𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
123, 11sylan 583 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1310, 12sseldi 3875 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝐷)
14 eqid 2738 . . . . . . . . . . . 12 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} = {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
151, 14psrbagconf1o 20749 . . . . . . . . . . 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 6618 . . . . . . . . . 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 6530 . . . . . . . 8 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
203adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑆) → 𝐹𝐷)
2120adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹𝐷)
221psrbagf 20731 . . . . . . . . . . . . . . . 16 (𝐹𝐷𝐹:𝐼⟶ℕ0)
2321, 22syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹:𝐼⟶ℕ0)
2423ffvelrnda 6861 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
25 simplr 769 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝑆)
2610, 25sseldi 3875 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝐷)
271psrbagf 20731 . . . . . . . . . . . . . . . 16 (𝑗𝐷𝑗:𝐼⟶ℕ0)
2826, 27syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗:𝐼⟶ℕ0)
2928ffvelrnda 6861 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
30 ssrab2 3969 . . . . . . . . . . . . . . . . 17 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ⊆ 𝐷
31 simpr 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
3230, 31sseldi 3875 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚𝐷)
331psrbagf 20731 . . . . . . . . . . . . . . . 16 (𝑚𝐷𝑚:𝐼⟶ℕ0)
3432, 33syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚:𝐼⟶ℕ0)
3534ffvelrnda 6861 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
36 nn0cn 11986 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
37 nn0cn 11986 . . . . . . . . . . . . . . 15 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
38 nn0cn 11986 . . . . . . . . . . . . . . 15 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
39 sub32 10998 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4036, 37, 38, 39syl3an 1161 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4124, 29, 35, 40syl3anc 1372 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4241mpteq2dva 5125 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
4334ffnd 6505 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 Fn 𝐼)
4431, 43fndmexd 7637 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐼 ∈ V)
45 ovexd 7205 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
4623feqmptd 6737 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
4728feqmptd 6737 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
4844, 24, 29, 46, 47offval2 7444 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
4934feqmptd 6737 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5044, 45, 35, 48, 49offval2 7444 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
51 ovexd 7205 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5244, 24, 35, 46, 49offval2 7444 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5344, 51, 29, 52, 47offval2 7444 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5442, 50, 533eqtr4d 2783 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = ((𝐹f𝑚) ∘f𝑗))
551, 14psrbagconcl 20747 . . . . . . . . . . . 12 (((𝐹f𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5613, 55sylan 583 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5754, 56eqeltrrd 2834 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
5854mpteq2dva 5125 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗)))
59 nfcv 2899 . . . . . . . . . . . 12 𝑛𝑋
60 nfcsb1v 3814 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝑋
61 csbeq1a 3804 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6259, 60, 61cbvmpt 5131 . . . . . . . . . . 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 3793 . . . . . . . . . 10 (𝑛 = ((𝐹f𝑚) ∘f𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
6557, 58, 63, 64fmptco 6901 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
6665feq1d 6489 . . . . . . . 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 235 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
6867fvmptelrn 6887 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
6968anasss 470 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
706, 69syldan 594 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
711, 2, 3, 4, 5, 70gsumbagdiag 20755 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
72 eqid 2738 . . . 4 (0g𝐺) = (0g𝐺)
731psrbaglefi 20745 . . . . . 6 (𝐹𝐷 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
743, 73syl 17 . . . . 5 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
752, 74eqeltrid 2837 . . . 4 (𝜑𝑆 ∈ Fin)
761, 2psrbagconcl 20747 . . . . . . 7 ((𝐹𝐷𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
773, 76sylan 583 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
7810, 77sseldi 3875 . . . . 5 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝐷)
791psrbaglefi 20745 . . . . 5 ((𝐹f𝑚) ∈ 𝐷 → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
8078, 79syl 17 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
81 xpfi 8863 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
8275, 75, 81syl2anc 587 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
83 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚𝑆)
846simpld 498 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑗𝑆)
85 brxp 5572 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
8683, 84, 85sylanbrc 586 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
8786pm2.24d 154 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
8887impr 458 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
894, 72, 5, 75, 80, 70, 82, 88gsum2d2 19213 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
901psrbaglefi 20745 . . . . 5 ((𝐹f𝑗) ∈ 𝐷 → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
9113, 90syl 17 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
92 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗𝑆)
931, 2, 3gsumbagdiaglem 20754 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}))
9493simpld 498 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑚𝑆)
95 brxp 5572 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
9692, 94, 95sylanbrc 586 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
9796pm2.24d 154 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
9897impr 458 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
994, 72, 5, 75, 91, 69, 82, 98gsum2d2 19213 . . 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 2781 . 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 484 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
10270anassrs 471 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
103102fmpttd 6889 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑚)}⟶𝐵)
104 ovex 7203 . . . . . . . . . . . 12 (ℕ0m 𝐼) ∈ V
1051, 104rabex2 5202 . . . . . . . . . . 11 𝐷 ∈ V
106105a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
107 rabexg 5199 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V)
108 mptexg 6994 . . . . . . . . . 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 6377 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
111110a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
112 fvexd 6689 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
113 suppssdm 7872 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
114 eqid 2738 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
115114dmmptss 6073 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
116113, 115sstri 3886 . . . . . . . . . 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 8921 . . . . . . . . 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 1379 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1204, 72, 101, 80, 103, 119gsumcl 19154 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) ∈ 𝐵)
121120fmpttd 6889 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))):𝑆𝐵)
1221, 2psrbagconf1o 20749 . . . . . . . 8 (𝐹𝐷 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
1233, 122syl 17 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
124 f1ocnv 6630 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
125 f1of 6618 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
126123, 124, 1253syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
127121, 126fcod 6530 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵)
128 coass 6098 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
129 f1ococnv2 6644 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
130123, 129syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
131130coeq2d 5705 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
132128, 131syl5eq 2785 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
133 eqidd 2739 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)) = (𝑚𝑆 ↦ (𝐹f𝑚)))
134 eqidd 2739 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
135 breq2 5034 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑥r𝑛𝑥r ≤ (𝐹f𝑚)))
136135rabbidv 3381 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → {𝑥𝐷𝑥r𝑛} = {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
137 ovex 7203 . . . . . . . . . . . . 13 (𝑛f𝑗) ∈ V
138 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)
139137, 138csbie 3825 . . . . . . . . . . . 12 (𝑛f𝑗) / 𝑘𝑋 = 𝑌
140 oveq1 7177 . . . . . . . . . . . . 13 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) = ((𝐹f𝑚) ∘f𝑗))
141140csbeq1d 3794 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
142139, 141eqtr3id 2787 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → 𝑌 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
143136, 142mpteq12dv 5115 . . . . . . . . . 10 (𝑛 = (𝐹f𝑚) → (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
144143oveq2d 7186 . . . . . . . . 9 (𝑛 = (𝐹f𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
14577, 133, 134, 144fmptco 6901 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
146145coeq1d 5704 . . . . . . 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 6097 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆)
148 ssid 3899 . . . . . . . . . 10 𝑆𝑆
149 resmpt 5879 . . . . . . . . . 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 2761 . . . . . . . 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 2781 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
154153feq1d 6489 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵))
155127, 154mpbid 235 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵)
156 rabexg 5199 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦r𝐹} ∈ V)
157105, 156mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ V)
1582, 157eqeltrid 2837 . . . . . 6 (𝜑𝑆 ∈ V)
159158mptexd 6997 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∈ V)
160 funmpt 6377 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
161160a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
162 fvexd 6689 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
163 suppssdm 7872 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
164 eqid 2738 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
165164dmmptss 6073 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ⊆ 𝑆
166163, 165sstri 3886 . . . . . 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 8921 . . . . 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 1379 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1704, 72, 5, 75, 155, 169, 123gsumf1o 19155 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))))
171145oveq2d 7186 . . 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 2773 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
1735adantr 484 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
174105a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
175 rabexg 5199 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V)
176 mptexg 6994 . . . . . . . 8 ({𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
177174, 175, 1763syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
178 funmpt 6377 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
179178a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))
180 fvexd 6689 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
181 suppssdm 7872 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
182 eqid 2738 . . . . . . . . . 10 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
183182dmmptss 6073 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
184181, 183sstri 3886 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
185184a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
186 suppssfifsupp 8921 . . . . . . 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 1379 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) finSupp (0g𝐺))
1884, 72, 173, 91, 9, 187, 16gsumf1o 19155 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))))
18965oveq2d 7186 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
190188, 189eqtrd 2773 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
191190mpteq2dva 5125 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
192191oveq2d 7186 . 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 2783 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 399   = wceq 1542  wcel 2114  {crab 3057  Vcvv 3398  csb 3790  wss 3843   class class class wbr 5030  cmpt 5110   I cid 5428   × cxp 5523  ccnv 5524  dom cdm 5525  cres 5527  cima 5528  ccom 5529  Fun wfun 6333  wf 6335  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  cmpo 7172  f cof 7423  r cofr 7424   supp csupp 7856  m cmap 8437  Fincfn 8555   finSupp cfsupp 8906  cc 10613  cle 10754  cmin 10948  cn 11716  0cn0 11976  Basecbs 16586  0gc0g 16816   Σg cgsu 16817  CMndccmn 19024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-ofr 7426  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-fzo 13125  df-seq 13461  df-hash 13783  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-0g 16818  df-gsum 16819  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-submnd 18073  df-mulg 18343  df-cntz 18565  df-cmn 19026
This theorem is referenced by:  psrass1  20784
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