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Theorem psrass1lem 19579
Description: A group sum commutation used by psrass1 19607. (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 19577 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}))
8 gsumbagdiag.x . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑋𝐵)
98anassrs 683 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑋𝐵)
10 eqid 2760 . . . . . . . . . . 11 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
119, 10fmptd 6548 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
123adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → 𝐼𝑉)
13 ssrab2 3828 . . . . . . . . . . . . . 14 {𝑦𝐷𝑦𝑟𝐹} ⊆ 𝐷
142, 13eqsstri 3776 . . . . . . . . . . . . 13 𝑆𝐷
154adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝐹𝐷)
16 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝑗𝑆)
171, 2psrbagconcl 19575 . . . . . . . . . . . . . 14 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1812, 15, 16, 17syl3anc 1477 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1914, 18sseldi 3742 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝐷)
20 eqid 2760 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
211, 20psrbagconf1o 19576 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2212, 19, 21syl2anc 696 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
23 f1of 6298 . . . . . . . . . . 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 6219 . . . . . . . . . 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 696 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
2712adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐼𝑉)
2815adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹𝐷)
291psrbagf 19567 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
3027, 28, 29syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹:𝐼⟶ℕ0)
3130ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3216adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝑆)
3314, 32sseldi 3742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝐷)
341psrbagf 19567 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3527, 33, 34syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗:𝐼⟶ℕ0)
3635ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
37 ssrab2 3828 . . . . . . . . . . . . . . . . . 18 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ⊆ 𝐷
38 simpr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
3937, 38sseldi 3742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚𝐷)
401psrbagf 19567 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
4127, 39, 40syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚:𝐼⟶ℕ0)
4241ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
43 nn0cn 11494 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
44 nn0cn 11494 . . . . . . . . . . . . . . . 16 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
45 nn0cn 11494 . . . . . . . . . . . . . . . 16 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
46 sub32 10507 . . . . . . . . . . . . . . . 16 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4743, 44, 45, 46syl3an 1164 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4831, 36, 42, 47syl3anc 1477 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4948mpteq2dva 4896 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
50 ovexd 6843 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
5130feqmptd 6411 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5235feqmptd 6411 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5327, 31, 36, 51, 52offval2 7079 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5441feqmptd 6411 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5527, 50, 42, 53, 54offval2 7079 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
56 ovexd 6843 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5727, 31, 42, 51, 54offval2 7079 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
5827, 56, 36, 57, 52offval2 7079 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
5949, 55, 583eqtr4d 2804 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
6019adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) ∈ 𝐷)
611, 20psrbagconcl 19575 . . . . . . . . . . . . 13 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6227, 60, 38, 61syl3anc 1477 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6359, 62eqeltrrd 2840 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6459mpteq2dva 4896 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗)))
65 nfcv 2902 . . . . . . . . . . . . 13 𝑛𝑋
66 nfcsb1v 3690 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑋
67 csbeq1a 3683 . . . . . . . . . . . . 13 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
6865, 66, 67cbvmpt 4901 . . . . . . . . . . . 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 3677 . . . . . . . . . . 11 (𝑛 = ((𝐹𝑓𝑚) ∘𝑓𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7163, 64, 69, 70fmptco 6559 . . . . . . . . . 10 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
7271feq1d 6191 . . . . . . . . 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 2760 . . . . . . . . 9 (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7574fmpt 6544 . . . . . . . 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 3070 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7877anasss 682 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
797, 78syldan 488 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
801, 2, 3, 4, 5, 6, 79gsumbagdiag 19578 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
81 eqid 2760 . . . 4 (0g𝐺) = (0g𝐺)
821psrbaglefi 19574 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
833, 4, 82syl2anc 696 . . . . 5 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
842, 83syl5eqel 2843 . . . 4 (𝜑𝑆 ∈ Fin)
853adantr 472 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
864adantr 472 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
87 simpr 479 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
881, 2psrbagconcl 19575 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
8985, 86, 87, 88syl3anc 1477 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
9014, 89sseldi 3742 . . . . 5 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝐷)
911psrbaglefi 19574 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑚) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
9285, 90, 91syl2anc 696 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
93 xpfi 8396 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
9484, 84, 93syl2anc 696 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
95 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚𝑆)
967simpld 477 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑗𝑆)
97 brxp 5304 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
9895, 96, 97sylanbrc 701 . . . . . 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 650 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1015, 81, 6, 84, 92, 79, 94, 100gsum2d2 18573 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1021psrbaglefi 19574 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
10312, 19, 102syl2anc 696 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
104 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗𝑆)
1051, 2, 3, 4gsumbagdiaglem 19577 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}))
106105simpld 477 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑚𝑆)
107 brxp 5304 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
108104, 106, 107sylanbrc 701 . . . . . 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 650 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1115, 81, 6, 84, 103, 78, 94, 110gsum2d2 18573 . . 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 2802 . 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 472 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
11479anassrs 683 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
115 eqid 2760 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
116114, 115fmptd 6548 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}⟶𝐵)
117 ovex 6841 . . . . . . . . . . . 12 (ℕ0𝑚 𝐼) ∈ V
1181, 117rabex2 4966 . . . . . . . . . . 11 𝐷 ∈ V
119118a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
120 rabexg 4963 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V)
121 mptexg 6648 . . . . . . . . . 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 6087 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
125 fvexd 6364 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
126 suppssdm 7476 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
127115dmmptss 5792 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
128126, 127sstri 3753 . . . . . . . . . 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 8455 . . . . . . . . 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 1485 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1325, 81, 113, 92, 116, 131gsumcl 18516 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) ∈ 𝐵)
133 eqid 2760 . . . . . . 7 (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
134132, 133fmptd 6548 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))):𝑆𝐵)
1351, 2psrbagconf1o 19576 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
1363, 4, 135syl2anc 696 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
137 f1ocnv 6310 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
138 f1of 6298 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
139136, 137, 1383syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
140 fco 6219 . . . . . 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 696 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵)
142 coass 5815 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
143 f1ococnv2 6324 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
144136, 143syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
145144coeq2d 5440 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
146142, 145syl5eq 2806 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
147 eqidd 2761 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)) = (𝑚𝑆 ↦ (𝐹𝑓𝑚)))
148 eqidd 2761 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
149 breq2 4808 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑥𝑟𝑛𝑥𝑟 ≤ (𝐹𝑓𝑚)))
150149rabbidv 3329 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → {𝑥𝐷𝑥𝑟𝑛} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
151 ovex 6841 . . . . . . . . . . . . 13 (𝑛𝑓𝑗) ∈ V
152 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛𝑓𝑗) → 𝑋 = 𝑌)
153151, 152csbie 3700 . . . . . . . . . . . 12 (𝑛𝑓𝑗) / 𝑘𝑋 = 𝑌
154 oveq1 6820 . . . . . . . . . . . . 13 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
155154csbeq1d 3681 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
156153, 155syl5eqr 2808 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → 𝑌 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
157150, 156mpteq12dv 4885 . . . . . . . . . 10 (𝑛 = (𝐹𝑓𝑚) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
158157oveq2d 6829 . . . . . . . . 9 (𝑛 = (𝐹𝑓𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
15989, 147, 148, 158fmptco 6559 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
160159coeq1d 5439 . . . . . . 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 5814 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆)
162 ssid 3765 . . . . . . . . . 10 𝑆𝑆
163 resmpt 5607 . . . . . . . . . 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 2782 . . . . . . . 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 2802 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
168167feq1d 6191 . . . . 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 4963 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
171118, 170mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
1722, 171syl5eqel 2843 . . . . . 6 (𝜑𝑆 ∈ V)
173 mptexg 6648 . . . . . 6 (𝑆 ∈ V → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
174172, 173syl 17 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
175 funmpt 6087 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
176175a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
177 fvexd 6364 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
178 suppssdm 7476 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
179 eqid 2760 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
180179dmmptss 5792 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ⊆ 𝑆
181178, 180sstri 3753 . . . . . 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 8455 . . . . 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 1485 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1855, 81, 6, 84, 169, 184, 136gsumf1o 18517 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))))
186159oveq2d 6829 . . 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 2794 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1886adantr 472 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
189118a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
190 rabexg 4963 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V)
191 mptexg 6648 . . . . . . . 8 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
192189, 190, 1913syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
193 funmpt 6087 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
194193a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))
195 fvexd 6364 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
196 suppssdm 7476 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
19710dmmptss 5792 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
198196, 197sstri 3753 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
199198a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
200 suppssfifsupp 8455 . . . . . . 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 1485 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) finSupp (0g𝐺))
2025, 81, 188, 103, 11, 201, 22gsumf1o 18517 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))))
20371oveq2d 6829 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
204202, 203eqtrd 2794 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
205204mpteq2dva 4896 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
206205oveq2d 6829 . 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 2804 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 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  csb 3674  wss 3715   class class class wbr 4804  cmpt 4881   I cid 5173   × cxp 5264  ccnv 5265  dom cdm 5266  cres 5268  cima 5269  ccom 5270  Fun wfun 6043  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑓 cof 7060  𝑟 cofr 7061   supp csupp 7463  𝑚 cmap 8023  Fincfn 8121   finSupp cfsupp 8440  cc 10126  cle 10267  cmin 10458  cn 11212  0cn0 11484  Basecbs 16059  0gc0g 16302   Σg cgsu 16303  CMndccmn 18393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-gsum 16305  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395
This theorem is referenced by:  psrass1  19607
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