Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  psrass1lem Structured version   Visualization version   GIF version

Theorem psrass1lem 19198
 Description: A group sum commutation used by psrass1 19226. (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 19196 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}))
8 gsumbagdiag.x . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑋𝐵)
98anassrs 678 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑋𝐵)
10 eqid 2610 . . . . . . . . . . 11 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
119, 10fmptd 6292 . . . . . . . . . 10 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
123adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → 𝐼𝑉)
13 ssrab2 3650 . . . . . . . . . . . . . 14 {𝑦𝐷𝑦𝑟𝐹} ⊆ 𝐷
142, 13eqsstri 3598 . . . . . . . . . . . . 13 𝑆𝐷
154adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝐹𝐷)
16 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑆) → 𝑗𝑆)
171, 2psrbagconcl 19194 . . . . . . . . . . . . . 14 ((𝐼𝑉𝐹𝐷𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1812, 15, 16, 17syl3anc 1318 . . . . . . . . . . . . 13 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝑆)
1914, 18sseldi 3566 . . . . . . . . . . . 12 ((𝜑𝑗𝑆) → (𝐹𝑓𝑗) ∈ 𝐷)
20 eqid 2610 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
211, 20psrbagconf1o 19195 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
2212, 19, 21syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}–1-1-onto→{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
23 f1of 6050 . . . . . . . . . . 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 5971 . . . . . . . . . 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 691 . . . . . . . . 9 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
2712adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐼𝑉)
2815adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹𝐷)
291psrbagf 19186 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
3027, 28, 29syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹:𝐼⟶ℕ0)
3130ffvelrnda 6267 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
3216adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝑆)
3314, 32sseldi 3566 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗𝐷)
341psrbagf 19186 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
3527, 33, 34syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗:𝐼⟶ℕ0)
3635ffvelrnda 6267 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
37 ssrab2 3650 . . . . . . . . . . . . . . . . . 18 {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ⊆ 𝐷
38 simpr 476 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
3937, 38sseldi 3566 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚𝐷)
401psrbagf 19186 . . . . . . . . . . . . . . . . 17 ((𝐼𝑉𝑚𝐷) → 𝑚:𝐼⟶ℕ0)
4127, 39, 40syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚:𝐼⟶ℕ0)
4241ffvelrnda 6267 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (𝑚𝑧) ∈ ℕ0)
43 nn0cn 11179 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℂ)
44 nn0cn 11179 . . . . . . . . . . . . . . . 16 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
45 nn0cn 11179 . . . . . . . . . . . . . . . 16 ((𝑚𝑧) ∈ ℕ0 → (𝑚𝑧) ∈ ℂ)
46 sub32 10194 . . . . . . . . . . . . . . . 16 (((𝐹𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ ∧ (𝑚𝑧) ∈ ℂ) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4743, 44, 45, 46syl3an 1360 . . . . . . . . . . . . . . 15 (((𝐹𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0 ∧ (𝑚𝑧) ∈ ℕ0) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4831, 36, 42, 47syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧)) = (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧)))
4948mpteq2dva 4672 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
50 ovex 6577 . . . . . . . . . . . . . . 15 ((𝐹𝑧) − (𝑗𝑧)) ∈ V
5150a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑗𝑧)) ∈ V)
5230feqmptd 6159 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝐹 = (𝑧𝐼 ↦ (𝐹𝑧)))
5335feqmptd 6159 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
5427, 31, 36, 52, 53offval2 6812 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑗𝑧))))
5541feqmptd 6159 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → 𝑚 = (𝑧𝐼 ↦ (𝑚𝑧)))
5627, 51, 42, 54, 55offval2 6812 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑗𝑧)) − (𝑚𝑧))))
57 ovex 6577 . . . . . . . . . . . . . . 15 ((𝐹𝑧) − (𝑚𝑧)) ∈ V
5857a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑚𝑧)) ∈ V)
5927, 31, 42, 52, 55offval2 6812 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑚) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑚𝑧))))
6027, 58, 36, 59, 53offval2 6812 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) = (𝑧𝐼 ↦ (((𝐹𝑧) − (𝑚𝑧)) − (𝑗𝑧))))
6149, 56, 603eqtr4d 2654 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
6219adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → (𝐹𝑓𝑗) ∈ 𝐷)
631, 20psrbagconcl 19194 . . . . . . . . . . . . 13 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6427, 62, 38, 63syl3anc 1318 . . . . . . . . . . . 12 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑗) ∘𝑓𝑚) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6561, 64eqeltrrd 2689 . . . . . . . . . . 11 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
6661mpteq2dva 4672 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗)))
67 nfcv 2751 . . . . . . . . . . . . 13 𝑛𝑋
68 nfcsb1v 3515 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑋
69 csbeq1a 3508 . . . . . . . . . . . . 13 (𝑘 = 𝑛𝑋 = 𝑛 / 𝑘𝑋)
7067, 68, 69cbvmpt 4677 . . . . . . . . . . . 12 (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋)
7170a1i 11 . . . . . . . . . . 11 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) = (𝑛 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑛 / 𝑘𝑋))
72 csbeq1 3502 . . . . . . . . . . 11 (𝑛 = ((𝐹𝑓𝑚) ∘𝑓𝑗) → 𝑛 / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7365, 66, 71, 72fmptco 6303 . . . . . . . . . 10 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
7473feq1d 5943 . . . . . . . . 9 ((𝜑𝑗𝑆) → (((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚))):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵))
7526, 74mpbid 221 . . . . . . . 8 ((𝜑𝑗𝑆) → (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
76 eqid 2610 . . . . . . . . 9 (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
7776fmpt 6289 . . . . . . . 8 (∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵 ↔ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}⟶𝐵)
7875, 77sylibr 223 . . . . . . 7 ((𝜑𝑗𝑆) → ∀𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
7978r19.21bi 2916 . . . . . 6 (((𝜑𝑗𝑆) ∧ 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
8079anasss 677 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
817, 80syldan 486 . . . 4 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
821, 2, 3, 4, 5, 6, 81gsumbagdiag 19197 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
83 eqid 2610 . . . 4 (0g𝐺) = (0g𝐺)
841psrbaglefi 19193 . . . . . 6 ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
853, 4, 84syl2anc 691 . . . . 5 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ Fin)
862, 85syl5eqel 2692 . . . 4 (𝜑𝑆 ∈ Fin)
873adantr 480 . . . . 5 ((𝜑𝑚𝑆) → 𝐼𝑉)
884adantr 480 . . . . . . 7 ((𝜑𝑚𝑆) → 𝐹𝐷)
89 simpr 476 . . . . . . 7 ((𝜑𝑚𝑆) → 𝑚𝑆)
901, 2psrbagconcl 19194 . . . . . . 7 ((𝐼𝑉𝐹𝐷𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
9187, 88, 89, 90syl3anc 1318 . . . . . 6 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝑆)
9214, 91sseldi 3566 . . . . 5 ((𝜑𝑚𝑆) → (𝐹𝑓𝑚) ∈ 𝐷)
931psrbaglefi 19193 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑚) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
9487, 92, 93syl2anc 691 . . . 4 ((𝜑𝑚𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ Fin)
95 xpfi 8116 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin)
9686, 86, 95syl2anc 691 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ Fin)
97 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚𝑆)
987simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑗𝑆)
99 brxp 5071 . . . . . . 7 (𝑚(𝑆 × 𝑆)𝑗 ↔ (𝑚𝑆𝑗𝑆))
10097, 98, 99sylanbrc 695 . . . . . 6 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → 𝑚(𝑆 × 𝑆)𝑗)
101100pm2.24d 146 . . . . 5 ((𝜑 ∧ (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})) → (¬ 𝑚(𝑆 × 𝑆)𝑗((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
102101impr 647 . . . 4 ((𝜑 ∧ ((𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) ∧ ¬ 𝑚(𝑆 × 𝑆)𝑗)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1035, 83, 6, 86, 94, 81, 96, 102gsum2d2 18196 . . 3 (𝜑 → (𝐺 Σg (𝑚𝑆, 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1041psrbaglefi 19193 . . . . 5 ((𝐼𝑉 ∧ (𝐹𝑓𝑗) ∈ 𝐷) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
10512, 19, 104syl2anc 691 . . . 4 ((𝜑𝑗𝑆) → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ Fin)
106 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗𝑆)
1071, 2, 3, 4gsumbagdiaglem 19196 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (𝑚𝑆𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}))
108107simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑚𝑆)
109 brxp 5071 . . . . . . 7 (𝑗(𝑆 × 𝑆)𝑚 ↔ (𝑗𝑆𝑚𝑆))
110106, 108, 109sylanbrc 695 . . . . . 6 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → 𝑗(𝑆 × 𝑆)𝑚)
111110pm2.24d 146 . . . . 5 ((𝜑 ∧ (𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑚((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺)))
112111impr 647 . . . 4 ((𝜑 ∧ ((𝑗𝑆𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑚)) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋 = (0g𝐺))
1135, 83, 6, 86, 105, 80, 96, 112gsum2d2 18196 . . 3 (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
11482, 103, 1133eqtr3d 2652 . 2 (𝜑 → (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1156adantr 480 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐺 ∈ CMnd)
11681anassrs 678 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}) → ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋𝐵)
117 eqid 2610 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
118116, 117fmptd 6292 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋):{𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}⟶𝐵)
119 ovex 6577 . . . . . . . . . . . 12 (ℕ0𝑚 𝐼) ∈ V
1201, 119rabex2 4742 . . . . . . . . . . 11 𝐷 ∈ V
121120a1i 11 . . . . . . . . . 10 ((𝜑𝑚𝑆) → 𝐷 ∈ V)
122 rabexg 4739 . . . . . . . . . 10 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V)
123 mptexg 6389 . . . . . . . . . 10 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ∈ V → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
124121, 122, 1233syl 18 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ∈ V)
125 funmpt 5840 . . . . . . . . . 10 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
126125a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
127 fvex 6113 . . . . . . . . . 10 (0g𝐺) ∈ V
128127a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → (0g𝐺) ∈ V)
129 suppssdm 7195 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
130117dmmptss 5548 . . . . . . . . . . 11 dom (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
131129, 130sstri 3577 . . . . . . . . . 10 ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)}
132131a1i 11 . . . . . . . . 9 ((𝜑𝑚𝑆) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
133 suppssfifsupp 8173 . . . . . . . . 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))
134124, 126, 128, 94, 132, 133syl32anc 1326 . . . . . . . 8 ((𝜑𝑚𝑆) → (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋) finSupp (0g𝐺))
1355, 83, 115, 94, 118, 134gsumcl 18139 . . . . . . 7 ((𝜑𝑚𝑆) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)) ∈ 𝐵)
136 eqid 2610 . . . . . . 7 (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
137135, 136fmptd 6292 . . . . . 6 (𝜑 → (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))):𝑆𝐵)
1381, 2psrbagconf1o 19195 . . . . . . . 8 ((𝐼𝑉𝐹𝐷) → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
1393, 4, 138syl2anc 691 . . . . . . 7 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
140 f1ocnv 6062 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆)
141 f1of 6050 . . . . . . 7 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
142139, 140, 1413syl 18 . . . . . 6 (𝜑(𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆𝑆)
143 fco 5971 . . . . . 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)
144137, 142, 143syl2anc 691 . . . . 5 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵)
145 coass 5571 . . . . . . . 8 (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
146 f1ococnv2 6076 . . . . . . . . . 10 ((𝑚𝑆 ↦ (𝐹𝑓𝑚)):𝑆1-1-onto𝑆 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
147139, 146syl 17 . . . . . . . . 9 (𝜑 → ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ( I ↾ 𝑆))
148147coeq2d 5206 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ((𝑚𝑆 ↦ (𝐹𝑓𝑚)) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
149145, 148syl5eq 2656 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)))
150 eqidd 2611 . . . . . . . . 9 (𝜑 → (𝑚𝑆 ↦ (𝐹𝑓𝑚)) = (𝑚𝑆 ↦ (𝐹𝑓𝑚)))
151 eqidd 2611 . . . . . . . . 9 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
152 breq2 4587 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑥𝑟𝑛𝑥𝑟 ≤ (𝐹𝑓𝑚)))
153152rabbidv 3164 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → {𝑥𝐷𝑥𝑟𝑛} = {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)})
154 ovex 6577 . . . . . . . . . . . . 13 (𝑛𝑓𝑗) ∈ V
155 psrass1lem.y . . . . . . . . . . . . 13 (𝑘 = (𝑛𝑓𝑗) → 𝑋 = 𝑌)
156154, 155csbie 3525 . . . . . . . . . . . 12 (𝑛𝑓𝑗) / 𝑘𝑋 = 𝑌
157 oveq1 6556 . . . . . . . . . . . . 13 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) = ((𝐹𝑓𝑚) ∘𝑓𝑗))
158157csbeq1d 3506 . . . . . . . . . . . 12 (𝑛 = (𝐹𝑓𝑚) → (𝑛𝑓𝑗) / 𝑘𝑋 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
159156, 158syl5eqr 2658 . . . . . . . . . . 11 (𝑛 = (𝐹𝑓𝑚) → 𝑌 = ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)
160153, 159mpteq12dv 4663 . . . . . . . . . 10 (𝑛 = (𝐹𝑓𝑚) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌) = (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))
161160oveq2d 6565 . . . . . . . . 9 (𝑛 = (𝐹𝑓𝑚) → (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)) = (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
16291, 150, 151, 161fmptco 6303 . . . . . . . 8 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
163162coeq1d 5205 . . . . . . 7 (𝜑 → (((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))))
164 coires1 5570 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆)
165 ssid 3587 . . . . . . . . . 10 𝑆𝑆
166 resmpt 5369 . . . . . . . . . 10 (𝑆𝑆 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
167165, 166ax-mp 5 . . . . . . . . 9 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ↾ 𝑆) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
168164, 167eqtri 2632 . . . . . . . 8 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
169168a1i 11 . . . . . . 7 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ ( I ↾ 𝑆)) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
170149, 163, 1693eqtr3d 2652 . . . . . 6 (𝜑 → ((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
171170feq1d 5943 . . . . 5 (𝜑 → (((𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚))):𝑆𝐵 ↔ (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵))
172144, 171mpbid 221 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))):𝑆𝐵)
173 rabexg 4739 . . . . . . . 8 (𝐷 ∈ V → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
174120, 173mp1i 13 . . . . . . 7 (𝜑 → {𝑦𝐷𝑦𝑟𝐹} ∈ V)
1752, 174syl5eqel 2692 . . . . . 6 (𝜑𝑆 ∈ V)
176 mptexg 6389 . . . . . 6 (𝑆 ∈ V → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
177175, 176syl 17 . . . . 5 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∈ V)
178 funmpt 5840 . . . . . 6 Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
179178a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))))
180127a1i 11 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
181 suppssdm 7195 . . . . . . 7 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
182 eqid 2610 . . . . . . . 8 (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) = (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))
183182dmmptss 5548 . . . . . . 7 dom (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ⊆ 𝑆
184181, 183sstri 3577 . . . . . 6 ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆
185184a1i 11 . . . . 5 (𝜑 → ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) supp (0g𝐺)) ⊆ 𝑆)
186 suppssfifsupp 8173 . . . . 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))
187177, 179, 180, 86, 185, 186syl32anc 1326 . . . 4 (𝜑 → (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) finSupp (0g𝐺))
1885, 83, 6, 86, 172, 187, 139gsumf1o 18140 . . 3 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))))
189162oveq2d 6565 . . 3 (𝜑 → (𝐺 Σg ((𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌))) ∘ (𝑚𝑆 ↦ (𝐹𝑓𝑚)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
190188, 189eqtrd 2644 . 2 (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑚𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑚)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
1916adantr 480 . . . . . 6 ((𝜑𝑗𝑆) → 𝐺 ∈ CMnd)
192120a1i 11 . . . . . . . 8 ((𝜑𝑗𝑆) → 𝐷 ∈ V)
193 rabexg 4739 . . . . . . . 8 (𝐷 ∈ V → {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V)
194 mptexg 6389 . . . . . . . 8 ({𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ∈ V → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
195192, 193, 1943syl 18 . . . . . . 7 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∈ V)
196 funmpt 5840 . . . . . . . 8 Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
197196a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → Fun (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))
198127a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → (0g𝐺) ∈ V)
199 suppssdm 7195 . . . . . . . . 9 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)
20010dmmptss 5548 . . . . . . . . 9 dom (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
201199, 200sstri 3577 . . . . . . . 8 ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)}
202201a1i 11 . . . . . . 7 ((𝜑𝑗𝑆) → ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) supp (0g𝐺)) ⊆ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)})
203 suppssfifsupp 8173 . . . . . . 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))
204195, 197, 198, 105, 202, 203syl32anc 1326 . . . . . 6 ((𝜑𝑗𝑆) → (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) finSupp (0g𝐺))
2055, 83, 191, 105, 11, 204, 22gsumf1o 18140 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))))
20673oveq2d 6565 . . . . 5 ((𝜑𝑗𝑆) → (𝐺 Σg ((𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋) ∘ (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑗) ∘𝑓𝑚)))) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
207205, 206eqtrd 2644 . . . 4 ((𝜑𝑗𝑆) → (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))
208207mpteq2dva 4672 . . 3 (𝜑 → (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋))) = (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋))))
209208oveq2d 6565 . 2 (𝜑 → (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ 𝑋)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑚 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑗)} ↦ ((𝐹𝑓𝑚) ∘𝑓𝑗) / 𝑘𝑋)))))
210114, 190, 2093eqtr4d 2654 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 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⦋csb 3499   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ∘𝑓 cof 6793   ∘𝑟 cofr 6794   supp csupp 7182   ↑𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  ℂcc 9813   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  0gc0g 15923   Σg cgsu 15924  CMndccmn 18016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018 This theorem is referenced by:  psrass1  19226
 Copyright terms: Public domain W3C validator