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Theorem gsumpart 30743
Description: Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumpart.b 𝐵 = (Base‘𝐺)
gsumpart.z 0 = (0g𝐺)
gsumpart.g (𝜑𝐺 ∈ CMnd)
gsumpart.a (𝜑𝐴𝑉)
gsumpart.x (𝜑𝑋𝑊)
gsumpart.f (𝜑𝐹:𝐴𝐵)
gsumpart.w (𝜑𝐹 finSupp 0 )
gsumpart.1 (𝜑Disj 𝑥𝑋 𝐶)
gsumpart.2 (𝜑 𝑥𝑋 𝐶 = 𝐴)
Assertion
Ref Expression
gsumpart (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)   0 (𝑥)

Proof of Theorem gsumpart
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumpart.b . . 3 𝐵 = (Base‘𝐺)
2 gsumpart.z . . 3 0 = (0g𝐺)
3 gsumpart.g . . 3 (𝜑𝐺 ∈ CMnd)
4 gsumpart.a . . 3 (𝜑𝐴𝑉)
5 gsumpart.f . . 3 (𝜑𝐹:𝐴𝐵)
6 gsumpart.w . . 3 (𝜑𝐹 finSupp 0 )
7 eqid 2801 . . . 4 𝑥𝑋 ({𝑥} × 𝐶) = 𝑥𝑋 ({𝑥} × 𝐶)
8 gsumpart.x . . . 4 (𝜑𝑋𝑊)
9 gsumpart.1 . . . 4 (𝜑Disj 𝑥𝑋 𝐶)
10 gsumpart.2 . . . 4 (𝜑 𝑥𝑋 𝐶 = 𝐴)
117, 4, 8, 9, 102ndresdjuf1o 30415 . . 3 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1-onto𝐴)
121, 2, 3, 4, 5, 6, 11gsumf1o 19032 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))))
13 snex 5300 . . . . . . 7 {𝑥} ∈ V
1413a1i 11 . . . . . 6 ((𝜑𝑥𝑋) → {𝑥} ∈ V)
154adantr 484 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴𝑉)
16 ssidd 3941 . . . . . . . . . 10 (𝜑𝐴𝐴)
1710, 16eqsstrd 3956 . . . . . . . . 9 (𝜑 𝑥𝑋 𝐶𝐴)
18 iunss 4935 . . . . . . . . 9 ( 𝑥𝑋 𝐶𝐴 ↔ ∀𝑥𝑋 𝐶𝐴)
1917, 18sylib 221 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 𝐶𝐴)
2019r19.21bi 3176 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐶𝐴)
2115, 20ssexd 5195 . . . . . 6 ((𝜑𝑥𝑋) → 𝐶 ∈ V)
2214, 21xpexd 7458 . . . . 5 ((𝜑𝑥𝑋) → ({𝑥} × 𝐶) ∈ V)
2322ralrimiva 3152 . . . 4 (𝜑 → ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
24 iunexg 7650 . . . 4 ((𝑋𝑊 ∧ ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V) → 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
258, 23, 24syl2anc 587 . . 3 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
26 relxp 5541 . . . . . 6 Rel ({𝑥} × 𝐶)
2726a1i 11 . . . . 5 ((𝜑𝑥𝑋) → Rel ({𝑥} × 𝐶))
2827ralrimiva 3152 . . . 4 (𝜑 → ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
29 reliun 5657 . . . 4 (Rel 𝑥𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
3028, 29sylibr 237 . . 3 (𝜑 → Rel 𝑥𝑋 ({𝑥} × 𝐶))
31 dmiun 5750 . . . . . 6 dom 𝑥𝑋 ({𝑥} × 𝐶) = 𝑥𝑋 dom ({𝑥} × 𝐶)
32 dmxpss 5999 . . . . . . . 8 dom ({𝑥} × 𝐶) ⊆ {𝑥}
3332rgenw 3121 . . . . . . 7 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34 ss2iun 4902 . . . . . . 7 (∀𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥} → 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥})
3533, 34ax-mp 5 . . . . . 6 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥}
3631, 35eqsstri 3952 . . . . 5 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥}
37 iunid 4950 . . . . 5 𝑥𝑋 {𝑥} = 𝑋
3836, 37sseqtri 3954 . . . 4 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
3938a1i 11 . . 3 (𝜑 → dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40 fo2nd 7696 . . . . . . . 8 2nd :V–onto→V
41 fof 6569 . . . . . . . 8 (2nd :V–onto→V → 2nd :V⟶V)
4240, 41ax-mp 5 . . . . . . 7 2nd :V⟶V
43 ssv 3942 . . . . . . 7 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V
44 fssres 6522 . . . . . . 7 ((2nd :V⟶V ∧ 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V)
4542, 43, 44mp2an 691 . . . . . 6 (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V
46 ffn 6491 . . . . . 6 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶))
4745, 46mp1i 13 . . . . 5 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶))
48 djussxp2 30413 . . . . . . . 8 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝑥𝑋 𝐶)
49 imass2 5936 . . . . . . . 8 ( 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝑥𝑋 𝐶) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × 𝑥𝑋 𝐶)))
5048, 49ax-mp 5 . . . . . . 7 (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × 𝑥𝑋 𝐶))
51 ima0 5916 . . . . . . . . . . 11 (2nd “ ∅) = ∅
52 xpeq1 5537 . . . . . . . . . . . . 13 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = (∅ × 𝑥𝑋 𝐶))
53 0xp 5617 . . . . . . . . . . . . 13 (∅ × 𝑥𝑋 𝐶) = ∅
5452, 53eqtrdi 2852 . . . . . . . . . . . 12 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = ∅)
5554imaeq2d 5900 . . . . . . . . . . 11 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = (2nd “ ∅))
56 iuneq1 4900 . . . . . . . . . . . 12 (𝑋 = ∅ → 𝑥𝑋 𝐶 = 𝑥 ∈ ∅ 𝐶)
57 0iun 4952 . . . . . . . . . . . 12 𝑥 ∈ ∅ 𝐶 = ∅
5856, 57eqtrdi 2852 . . . . . . . . . . 11 (𝑋 = ∅ → 𝑥𝑋 𝐶 = ∅)
5951, 55, 583eqtr4a 2862 . . . . . . . . . 10 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6059adantl 485 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
61 2ndimaxp 30412 . . . . . . . . . 10 (𝑋 ≠ ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6261adantl 485 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6360, 62pm2.61dane 3077 . . . . . . . 8 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6463, 10eqtrd 2836 . . . . . . 7 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝐴)
6550, 64sseqtrid 3970 . . . . . 6 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66 resssxp 6093 . . . . . 6 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴 ↔ (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴))
6765, 66sylib 221 . . . . 5 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴))
68 dff2 6846 . . . . 5 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴 ↔ ((2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶) ∧ (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴)))
6947, 67, 68sylanbrc 586 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
705, 69fcod 6510 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐵)
717, 4, 8, 9, 102ndresdju 30414 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1𝐴)
722fvexi 6663 . . . . 5 0 ∈ V
7372a1i 11 . . . 4 (𝜑0 ∈ V)
745, 4fexd 6971 . . . 4 (𝜑𝐹 ∈ V)
756, 71, 73, 74fsuppco 8853 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))) finSupp 0 )
761, 2, 3, 25, 30, 8, 39, 70, 75gsum2d 19088 . 2 (𝜑 → (𝐺 Σg (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))) = (𝐺 Σg (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))))))
77 nfcsb1v 3855 . . . . . . . . 9 𝑥𝑦 / 𝑥𝐶
78 csbeq1a 3845 . . . . . . . . 9 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
798, 21, 77, 78iunsnima2 30386 . . . . . . . 8 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) = 𝑦 / 𝑥𝐶)
80 df-ov 7142 . . . . . . . . 9 (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
8169ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
82 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦𝑋)
83 vsnid 4565 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑦}
8483a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
8579eleq2d 2878 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↔ 𝑧𝑦 / 𝑥𝐶))
8685biimpa 480 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑧𝑦 / 𝑥𝐶)
8784, 86opelxpd 5561 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶))
88 nfcv 2958 . . . . . . . . . . . . . . . 16 𝑥{𝑦}
8988, 77nfxp 5556 . . . . . . . . . . . . . . 15 𝑥({𝑦} × 𝑦 / 𝑥𝐶)
9089nfel2 2976 . . . . . . . . . . . . . 14 𝑥𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)
91 sneq 4538 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
9291, 78xpeq12d 5554 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × 𝑦 / 𝑥𝐶))
9392eleq2d 2878 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)))
9490, 93rspce 3563 . . . . . . . . . . . . 13 ((𝑦𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
9582, 87, 94syl2anc 587 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96 eliun 4888 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝑋 ({𝑥} × 𝐶) ↔ ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
9795, 96sylibr 237 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝑥𝑋 ({𝑥} × 𝐶))
9881, 97fvco3d 6742 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)))
9997fvresd 6669 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100 vex 3447 . . . . . . . . . . . . 13 𝑦 ∈ V
101 vex 3447 . . . . . . . . . . . . 13 𝑧 ∈ V
102100, 101op2nd 7684 . . . . . . . . . . . 12 (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
10399, 102eqtrdi 2852 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104103fveq2d 6653 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹𝑧))
10598, 104eqtrd 2836 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹𝑧))
10680, 105syl5eq 2848 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹𝑧))
10779, 106mpteq12dva 5117 . . . . . . 7 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
1085adantr 484 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝐹:𝐴𝐵)
109 imassrn 5911 . . . . . . . . . 10 ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran 𝑥𝑋 ({𝑥} × 𝐶)
11010xpeq2d 5553 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 × 𝑥𝑋 𝐶) = (𝑋 × 𝐴))
11148, 110sseqtrid 3970 . . . . . . . . . . . . 13 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112 rnss 5777 . . . . . . . . . . . . 13 ( 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
113111, 112syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
114113adantr 484 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115 rnxpss 6000 . . . . . . . . . . 11 ran (𝑋 × 𝐴) ⊆ 𝐴
116114, 115sstrdi 3930 . . . . . . . . . 10 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117109, 116sstrid 3929 . . . . . . . . 9 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
11879, 117eqsstrrd 3957 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐶𝐴)
119108, 118feqresmpt 6713 . . . . . . 7 ((𝜑𝑦𝑋) → (𝐹𝑦 / 𝑥𝐶) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
120107, 119eqtr4d 2839 . . . . . 6 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝐹𝑦 / 𝑥𝐶))
121120oveq2d 7155 . . . . 5 ((𝜑𝑦𝑋) → (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
122121mpteq2dva 5128 . . . 4 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶))))
123 nfcv 2958 . . . . 5 𝑦(𝐺 Σg (𝐹𝐶))
124 nfcv 2958 . . . . . 6 𝑥𝐺
125 nfcv 2958 . . . . . 6 𝑥 Σg
126 nfcv 2958 . . . . . . 7 𝑥𝐹
127126, 77nfres 5824 . . . . . 6 𝑥(𝐹𝑦 / 𝑥𝐶)
128124, 125, 127nfov 7169 . . . . 5 𝑥(𝐺 Σg (𝐹𝑦 / 𝑥𝐶))
12978reseq2d 5822 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝐶) = (𝐹𝑦 / 𝑥𝐶))
130129oveq2d 7155 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
131123, 128, 130cbvmpt 5134 . . . 4 (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
132122, 131eqtr4di 2854 . . 3 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))))
133132oveq2d 7155 . 2 (𝜑 → (𝐺 Σg (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))))) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
13412, 76, 1333eqtrd 2840 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  Vcvv 3444  csb 3831  wss 3884  c0 4246  {csn 4528  cop 4534   ciun 4884  Disj wdisj 4998   class class class wbr 5033  cmpt 5113   × cxp 5521  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  ccom 5527  Rel wrel 5528   Fn wfn 6323  wf 6324  ontowfo 6326  cfv 6328  (class class class)co 7139  2nd c2nd 7674   finSupp cfsupp 8821  Basecbs 16478  0gc0g 16708   Σg cgsu 16709  CMndccmn 18901
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903
This theorem is referenced by:  elrspunidl  31017
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