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Theorem psrass1lem 20078
Description: A group sum commutation used by psrass1 20106. (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 20076 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}))
8 gsumbagdiag.x . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)
98anassrs 468 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑋𝐵)
109fmpttd 6874 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
113adantr 481 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → 𝐼𝑉)
122ssrab3 4060 . . . . . . . . . . . 12 𝑆𝐷
134adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → 𝐹𝐷)
14 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → 𝑗𝑆)
151, 2psrbagconcl 20074 . . . . . . . . . . . . 13 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1611, 13, 14, 15syl3anc 1365 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1712, 16sseldi 3968 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝐷)
18 eqid 2824 . . . . . . . . . . . 12 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} = {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
191, 18psrbagconf1o 20075 . . . . . . . . . . 11 ((𝐼𝑉 ∧ (𝐹f𝑗) ∈ 𝐷) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
2011, 17, 19syl2anc 584 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
21 f1of 6611 . . . . . . . . . 10 ((𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}–1-1-onto→{𝑥𝐷𝑥r ≤ (𝐹f𝑗)} → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
2220, 21syl 17 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶{𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
23 fco 6527 . . . . . . . . 9 (((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)
2410, 22, 23syl2anc 584 . . . . . . . 8 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
2511adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐼𝑉)
2613adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹𝐷)
271psrbagf 20066 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
2825, 26, 27syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹:𝐼⟶ℕ0)
2928ffvelrnda 6846 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3014adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝑆)
3112, 30sseldi 3968 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝐷)
321psrbagf 20066 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3325, 31, 32syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗:𝐼⟶ℕ0)
3433ffvelrnda 6846 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
35 ssrab2 4059 . . . . . . . . . . . . . . . . 17 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ⊆ 𝐷
36 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
3735, 36sseldi 3968 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚𝐷)
381psrbagf 20066 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
3925, 37, 38syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚:𝐼⟶ℕ0)
4039ffvelrnda 6846 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
41 nn0cn 11899 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
42 nn0cn 11899 . . . . . . . . . . . . . . 15 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
43 nn0cn 11899 . . . . . . . . . . . . . . 15 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
44 sub32 10912 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4541, 42, 43, 44syl3an 1154 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4629, 34, 40, 45syl3anc 1365 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4746mpteq2dva 5157 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
48 ovexd 7186 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
4928feqmptd 6729 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5033feqmptd 6729 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5125, 29, 34, 49, 50offval2 7419 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5239feqmptd 6729 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5325, 48, 40, 51, 52offval2 7419 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
54 ovexd 7186 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5525, 29, 40, 49, 52offval2 7419 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5625, 54, 34, 55, 50offval2 7419 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5747, 53, 563eqtr4d 2870 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = ((𝐹f𝑚) ∘f𝑗))
5817adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) ∈ 𝐷)
591, 18psrbagconcl 20074 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹f𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6025, 58, 36, 59syl3anc 1365 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6157, 60eqeltrrd 2918 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6257mpteq2dva 5157 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗)))
63 nfcv 2981 . . . . . . . . . . . 12 𝑛𝑋
64 nfcsb1v 3910 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝑋
65 csbeq1a 3900 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6663, 64, 65cbvmpt 5163 . . . . . . . . . . 11 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑛 / 𝑘𝑋)
6766a1i 11 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑛 / 𝑘𝑋))
68 csbeq1 3889 . . . . . . . . . 10 (𝑛 = ((𝐹f𝑚) ∘f𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
6961, 62, 67, 68fmptco 6886 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
7069feq1d 6495 . . . . . . . 8 ((𝜑𝑗𝑆) → (((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵))
7124, 70mpbid 233 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
7271fvmptelrn 6872 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
7372anasss 467 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
747, 73syldan 591 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
751, 2, 3, 4, 5, 6, 74gsumbagdiag 20077 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
76 eqid 2824 . . . 4 (0g𝐺) = (0g𝐺)
771psrbaglefi 20073 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦r𝐹} ∈ Fin)
783, 4, 77syl2anc 584 . . . . 5 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
792, 78eqeltrid 2921 . . . 4 (𝜑𝑆 ∈ Fin)
803adantr 481 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
814adantr 481 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
82 simpr 485 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
831, 2psrbagconcl 20074 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
8480, 81, 82, 83syl3anc 1365 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
8512, 84sseldi 3968 . . . . 5 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝐷)
861psrbaglefi 20073 . . . . 5 ((𝐼𝑉 ∧ (𝐹f𝑚) ∈ 𝐷) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
8780, 85, 86syl2anc 584 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
88 xpfi 8781 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
8979, 79, 88syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
90 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚𝑆)
917simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑗𝑆)
92 brxp 5599 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
9390, 91, 92sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
9493pm2.24d 154 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
9594impr 455 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
965, 76, 6, 79, 87, 74, 89, 95gsum2d2 19016 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
971psrbaglefi 20073 . . . . 5 ((𝐼𝑉 ∧ (𝐹f𝑗) ∈ 𝐷) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
9811, 17, 97syl2anc 584 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
99 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗𝑆)
1001, 2, 3, 4gsumbagdiaglem 20076 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}))
101100simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑚𝑆)
102 brxp 5599 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
10399, 101, 102sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
104103pm2.24d 154 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺)))
105104impr 455 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋 = (0g𝐺))
1065, 76, 6, 79, 98, 73, 89, 105gsum2d2 19016 . . 3 (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
10775, 96, 1063eqtr3d 2868 . 2 (𝜑 → (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
1086adantr 481 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
10974anassrs 468 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
110109fmpttd 6874 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑚)}⟶𝐵)
111 ovex 7184 . . . . . . . . . . . 12 (ℕ0m 𝐼) ∈ V
1121, 111rabex2 5233 . . . . . . . . . . 11 𝐷 ∈ V
113112a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
114 rabexg 5230 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V)
115 mptexg 6982 . . . . . . . . . 10 ({𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ∈ V)
116113, 114, 1153syl 18 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ∈ V)
117 funmpt 6389 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
118117a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
119 fvexd 6681 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
120 suppssdm 7837 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
121 eqid 2824 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
122121dmmptss 6092 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
123120, 122sstri 3979 . . . . . . . . . 10 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
125 suppssfifsupp 8840 . . . . . . . . 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))
126116, 118, 119, 87, 124, 125syl32anc 1372 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1275, 76, 108, 87, 110, 126gsumcl 18957 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) ∈ 𝐵)
128127fmpttd 6874 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))):𝑆𝐵)
1291, 2psrbagconf1o 20075 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
1303, 4, 129syl2anc 584 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
131 f1ocnv 6623 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
132 f1of 6611 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
133130, 131, 1323syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
134 fco 6527 . . . . . 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)
135128, 133, 134syl2anc 584 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵)
136 coass 6115 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
137 f1ococnv2 6637 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
138130, 137syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
139138coeq2d 5731 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
140136, 139syl5eq 2872 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
141 eqidd 2825 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)) = (𝑚𝑆 ↦ (𝐹f𝑚)))
142 eqidd 2825 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
143 breq2 5066 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑥r𝑛𝑥r ≤ (𝐹f𝑚)))
144143rabbidv 3485 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → {𝑥𝐷𝑥r𝑛} = {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
145 ovex 7184 . . . . . . . . . . . . 13 (𝑛f𝑗) ∈ V
146 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)
147145, 146csbie 3921 . . . . . . . . . . . 12 (𝑛f𝑗) / 𝑘𝑋 = 𝑌
148 oveq1 7158 . . . . . . . . . . . . 13 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) = ((𝐹f𝑚) ∘f𝑗))
149148csbeq1d 3890 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
150147, 149syl5eqr 2874 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → 𝑌 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
151144, 150mpteq12dv 5147 . . . . . . . . . 10 (𝑛 = (𝐹f𝑚) → (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
152151oveq2d 7167 . . . . . . . . 9 (𝑛 = (𝐹f𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
15384, 141, 142, 152fmptco 6886 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
154153coeq1d 5730 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
155 coires1 6114 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆)
156 ssid 3992 . . . . . . . . . 10 𝑆𝑆
157 resmpt 5903 . . . . . . . . . 10 (𝑆𝑆 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
158156, 157ax-mp 5 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
159155, 158eqtri 2848 . . . . . . . 8 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
160159a1i 11 . . . . . . 7 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
161140, 154, 1603eqtr3d 2868 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
162161feq1d 6495 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵))
163135, 162mpbid 233 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵)
164 rabexg 5230 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦r𝐹} ∈ V)
165112, 164mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ V)
1662, 165eqeltrid 2921 . . . . . 6 (𝜑𝑆 ∈ V)
167166mptexd 6985 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∈ V)
168 funmpt 6389 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
169168a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
170 fvexd 6681 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
171 suppssdm 7837 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
172 eqid 2824 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
173172dmmptss 6092 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ⊆ 𝑆
174171, 173sstri 3979 . . . . . 6 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆
175174a1i 11 . . . . 5 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)
176 suppssfifsupp 8840 . . . . 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))
177167, 169, 170, 79, 175, 176syl32anc 1372 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1785, 76, 6, 79, 163, 177, 130gsumf1o 18958 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))))
179153oveq2d 7167 . . 3 (𝜑 → (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
180178, 179eqtrd 2860 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
1816adantr 481 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
182112a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
183 rabexg 5230 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V)
184 mptexg 6982 . . . . . . . 8 ({𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
185182, 183, 1843syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
186 funmpt 6389 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
187186a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))
188 fvexd 6681 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
189 suppssdm 7837 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
190 eqid 2824 . . . . . . . . . 10 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
191190dmmptss 6092 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
192189, 191sstri 3979 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
193192a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
194 suppssfifsupp 8840 . . . . . . 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))
195185, 187, 188, 98, 193, 194syl32anc 1372 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) finSupp (0g𝐺))
1965, 76, 181, 98, 10, 195, 20gsumf1o 18958 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))))
19769oveq2d 7167 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
198196, 197eqtrd 2860 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
199198mpteq2dva 5157 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
200199oveq2d 7167 . 2 (𝜑 → (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
201107, 180, 2003eqtr4d 2870 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 1530  wcel 2106  {crab 3146  Vcvv 3499  csb 3886  wss 3939   class class class wbr 5062  cmpt 5142   I cid 5457   × cxp 5551  ccnv 5552  dom cdm 5553  cres 5555  cima 5556  ccom 5557  Fun wfun 6345  wf 6347  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  cmpo 7153  f cof 7400  r cofr 7401   supp csupp 7824  m cmap 8399  Fincfn 8501   finSupp cfsupp 8825  cc 10527  cle 10668  cmin 10862  cn 11630  0cn0 11889  Basecbs 16475  0gc0g 16705   Σg cgsu 16706  CMndccmn 18828
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-ofr 7403  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947  df-mulg 18157  df-cntz 18379  df-cmn 18830
This theorem is referenced by:  psrass1  20106
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