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Theorem psrass1lemOLD 21342
Description: Obsolete version of psrass1lem 21345 as of 7-Aug-2024. (Contributed by Mario Carneiro, 5-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.s 𝑆 = {𝑦𝐷𝑦r𝐹}
gsumbagdiagOLD.i (𝜑𝐼𝑉)
gsumbagdiagOLD.f (𝜑𝐹𝐷)
gsumbagdiagOLD.b 𝐵 = (Base‘𝐺)
gsumbagdiagOLD.g (𝜑𝐺 ∈ CMnd)
gsumbagdiagOLD.x ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)
psrass1lemOLD.y (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)
Assertion
Ref Expression
psrass1lemOLD (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 psrass1lemOLD
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.s . . . 4 𝑆 = {𝑦𝐷𝑦r𝐹}
3 gsumbagdiagOLD.i . . . 4 (𝜑𝐼𝑉)
4 gsumbagdiagOLD.f . . . 4 (𝜑𝐹𝐷)
5 gsumbagdiagOLD.b . . . 4 𝐵 = (Base‘𝐺)
6 gsumbagdiagOLD.g . . . 4 (𝜑𝐺 ∈ CMnd)
71, 2, 3, 4gsumbagdiaglemOLD 21340 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}))
8 gsumbagdiagOLD.x . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)
98anassrs 468 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑋𝐵)
109fmpttd 7063 . . . . . . . . 9 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
113adantr 481 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → 𝐼𝑉)
122ssrab3 4040 . . . . . . . . . . . 12 𝑆𝐷
134adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → 𝐹𝐷)
14 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → 𝑗𝑆)
151, 2psrbagconclOLD 21337 . . . . . . . . . . . . 13 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1611, 13, 14, 15syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝑆)
1712, 16sselid 3942 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝐹f𝑗) ∈ 𝐷)
18 eqid 2736 . . . . . . . . . . . 12 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} = {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
191, 18psrbagconf1oOLD 21339 . . . . . . . . . . 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 6784 . . . . . . . . . 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 6692 . . . . . . . . 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)
271psrbagfOLD 21321 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
2825, 26, 27syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹:𝐼⟶ℕ0)
2928ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3014adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝑆)
3112, 30sselid 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗𝐷)
321psrbagfOLD 21321 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3325, 31, 32syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗:𝐼⟶ℕ0)
3433ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
35 ssrab2 4037 . . . . . . . . . . . . . . . . 17 {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ⊆ 𝐷
36 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
3735, 36sselid 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚𝐷)
381psrbagfOLD 21321 . . . . . . . . . . . . . . . 16 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
3925, 37, 38syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚:𝐼⟶ℕ0)
4039ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
41 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
42 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
43 nn0cn 12423 . . . . . . . . . . . . . . 15 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
44 sub32 11435 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4541, 42, 43, 44syl3an 1160 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4629, 34, 40, 45syl3anc 1371 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4746mpteq2dva 5205 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
48 ovexd 7392 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
4928feqmptd 6910 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5033feqmptd 6910 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5125, 29, 34, 49, 50offval2 7637 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5239feqmptd 6910 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5325, 48, 40, 51, 52offval2 7637 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
54 ovexd 7392 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5525, 29, 40, 49, 52offval2 7637 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5625, 54, 34, 55, 50offval2 7637 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5747, 53, 563eqtr4d 2786 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) = ((𝐹f𝑚) ∘f𝑗))
5817adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → (𝐹f𝑗) ∈ 𝐷)
591, 18psrbagconclOLD 21337 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹f𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6025, 58, 36, 59syl3anc 1371 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑗) ∘f𝑚) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6157, 60eqeltrrd 2839 . . . . . . . . . 10 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}) → ((𝐹f𝑚) ∘f𝑗) ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
6257mpteq2dva 5205 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗)))
63 nfcv 2907 . . . . . . . . . . . 12 𝑛𝑋
64 nfcsb1v 3880 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝑋
65 csbeq1a 3869 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6663, 64, 65cbvmpt 5216 . . . . . . . . . . 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 3858 . . . . . . . . . 10 (𝑛 = ((𝐹f𝑚) ∘f𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
6961, 62, 67, 68fmptco 7075 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
7069feq1d 6653 . . . . . . . 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 231 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑗)}⟶𝐵)
7271fvmptelcdm 7061 . . . . . 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, 74gsumbagdiagOLD 21341 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
76 eqid 2736 . . . 4 (0g𝐺) = (0g𝐺)
771psrbaglefiOLD 21335 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦r𝐹} ∈ Fin)
783, 4, 77syl2anc 584 . . . . 5 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ Fin)
792, 78eqeltrid 2842 . . . 4 (𝜑𝑆 ∈ Fin)
803adantr 481 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
814adantr 481 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
82 simpr 485 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
831, 2psrbagconclOLD 21337 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
8480, 81, 82, 83syl3anc 1371 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝑆)
8512, 84sselid 3942 . . . . 5 ((𝜑𝑚𝑆) → (𝐹f𝑚) ∈ 𝐷)
861psrbaglefiOLD 21335 . . . . 5 ((𝐼𝑉 ∧ (𝐹f𝑚) ∈ 𝐷) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
8780, 85, 86syl2anc 584 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ Fin)
88 xpfi 9261 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
8979, 79, 88syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
90 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚𝑆)
917simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑗𝑆)
92 brxp 5681 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
9390, 91, 92sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
9493pm2.24d 151 . . . . 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 19751 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))))
971psrbaglefiOLD 21335 . . . . 5 ((𝐼𝑉 ∧ (𝐹f𝑗) ∈ 𝐷) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
9811, 17, 97syl2anc 584 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ Fin)
99 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗𝑆)
1001, 2, 3, 4gsumbagdiaglemOLD 21340 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}))
101100simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑚𝑆)
102 brxp 5681 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
10399, 101, 102sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
104103pm2.24d 151 . . . . 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 19751 . . 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 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)))))
1086adantr 481 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
10974anassrs 468 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}) → ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋𝐵)
110109fmpttd 7063 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋):{𝑥𝐷𝑥r ≤ (𝐹f𝑚)}⟶𝐵)
111 ovex 7390 . . . . . . . . . . . 12 (ℕ0m 𝐼) ∈ V
1121, 111rabex2 5291 . . . . . . . . . . 11 𝐷 ∈ V
113112a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
114 rabexg 5288 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ∈ V)
115 mptexg 7171 . . . . . . . . . 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 6539 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
118117a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
119 fvexd 6857 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
120 suppssdm 8108 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
121 eqid 2736 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
122121dmmptss 6193 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)}
123120, 122sstri 3953 . . . . . . . . . 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 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))
126116, 118, 119, 87, 124, 125syl32anc 1378 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1275, 76, 108, 87, 110, 126gsumcl 19692 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)) ∈ 𝐵)
128127fmpttd 7063 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))):𝑆𝐵)
1291, 2psrbagconf1oOLD 21339 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
1303, 4, 129syl2anc 584 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
131 f1ocnv 6796 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆)
132 f1of 6784 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
133130, 131, 1323syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹f𝑚)):𝑆𝑆)
134 fco 6692 . . . . . 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 6217 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))))
137 f1ococnv2 6811 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹f𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
138130, 137syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ( I ↾ 𝑆))
139138coeq2d 5818 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹f𝑚)) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
140136, 139eqtrid 2788 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
141 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹f𝑚)) = (𝑚𝑆 ↦ (𝐹f𝑚)))
142 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
143 breq2 5109 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑥r𝑛𝑥r ≤ (𝐹f𝑚)))
144143rabbidv 3415 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → {𝑥𝐷𝑥r𝑛} = {𝑥𝐷𝑥r ≤ (𝐹f𝑚)})
145 ovex 7390 . . . . . . . . . . . . 13 (𝑛f𝑗) ∈ V
146 psrass1lemOLD.y . . . . . . . . . . . . 13 (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)
147145, 146csbie 3891 . . . . . . . . . . . 12 (𝑛f𝑗) / 𝑘𝑋 = 𝑌
148 oveq1 7364 . . . . . . . . . . . . 13 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) = ((𝐹f𝑚) ∘f𝑗))
149148csbeq1d 3859 . . . . . . . . . . . 12 (𝑛 = (𝐹f𝑚) → (𝑛f𝑗) / 𝑘𝑋 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
150147, 149eqtr3id 2790 . . . . . . . . . . 11 (𝑛 = (𝐹f𝑚) → 𝑌 = ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)
151144, 150mpteq12dv 5196 . . . . . . . . . 10 (𝑛 = (𝐹f𝑚) → (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))
152151oveq2d 7373 . . . . . . . . 9 (𝑛 = (𝐹f𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
15384, 141, 142, 152fmptco 7075 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
154153coeq1d 5817 . . . . . . 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 6216 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ↾ 𝑆)
156 ssid 3966 . . . . . . . . . 10 𝑆𝑆
157 resmpt 5991 . . . . . . . . . 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 2764 . . . . . . . 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 2784 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
162161feq1d 6653 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑚)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵))
163135, 162mpbid 231 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))):𝑆𝐵)
164 rabexg 5288 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦r𝐹} ∈ V)
165112, 164mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦r𝐹} ∈ V)
1662, 165eqeltrid 2842 . . . . . 6 (𝜑𝑆 ∈ V)
167166mptexd 7174 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∈ V)
168 funmpt 6539 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
169168a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))))
170 fvexd 6857 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
171 suppssdm 8108 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
172 eqid 2736 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))
173172dmmptss 6193 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ⊆ 𝑆
174171, 173sstri 3953 . . . . . 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 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))
177167, 169, 170, 79, 175, 176syl32anc 1378 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1785, 76, 6, 79, 163, 177, 130gsumf1o 19693 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹f𝑚)))))
179153oveq2d 7373 . . 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 2776 . 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 5288 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V)
184 mptexg 7171 . . . . . . . 8 ({𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
185182, 183, 1843syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∈ V)
186 funmpt 6539 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
187186a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))
188 fvexd 6857 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
189 suppssdm 8108 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
190 eqid 2736 . . . . . . . . . 10 (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)
191190dmmptss 6193 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
192189, 191sstri 3953 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)}
193192a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})
194 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))
195185, 187, 188, 98, 193, 194syl32anc 1378 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) finSupp (0g𝐺))
1965, 76, 181, 98, 10, 195, 20gsumf1o 19693 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))))
19769oveq2d 7373 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑗) ∘f𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
198196, 197eqtrd 2776 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋)))
199198mpteq2dva 5205 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ ((𝐹f𝑚) ∘f𝑗) / 𝑘𝑋))))
200199oveq2d 7373 . 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 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: (None)
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