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Theorem gsumcom2 18314
Description: Two-dimensional commutation of a group sum. Note that while 𝐴 and 𝐷 are constants w.r.t. 𝑗, 𝑘, 𝐶(𝑗) and 𝐸(𝑘) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
gsum2d2.b 𝐵 = (Base‘𝐺)
gsum2d2.z 0 = (0g𝐺)
gsum2d2.g (𝜑𝐺 ∈ CMnd)
gsum2d2.a (𝜑𝐴𝑉)
gsum2d2.r ((𝜑𝑗𝐴) → 𝐶𝑊)
gsum2d2.f ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)
gsum2d2.u (𝜑𝑈 ∈ Fin)
gsum2d2.n ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
gsumcom2.d (𝜑𝐷𝑌)
gsumcom2.c (𝜑 → ((𝑗𝐴𝑘𝐶) ↔ (𝑘𝐷𝑗𝐸)))
Assertion
Ref Expression
gsumcom2 (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐷, 𝑗𝐸𝑋)))
Distinct variable groups:   𝑗,𝑘,𝐵   𝐷,𝑗,𝑘   𝑗,𝐸   𝜑,𝑗,𝑘   𝐴,𝑗,𝑘   𝑗,𝐺,𝑘   𝑈,𝑗,𝑘   𝐶,𝑘   𝑗,𝑉   0 ,𝑗,𝑘
Allowed substitution hints:   𝐶(𝑗)   𝐸(𝑘)   𝑉(𝑘)   𝑊(𝑗,𝑘)   𝑋(𝑗,𝑘)   𝑌(𝑗,𝑘)

Proof of Theorem gsumcom2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsum2d2.b . . 3 𝐵 = (Base‘𝐺)
2 gsum2d2.z . . 3 0 = (0g𝐺)
3 gsum2d2.g . . 3 (𝜑𝐺 ∈ CMnd)
4 gsum2d2.a . . . 4 (𝜑𝐴𝑉)
5 snex 4879 . . . . . 6 {𝑗} ∈ V
6 gsum2d2.r . . . . . 6 ((𝜑𝑗𝐴) → 𝐶𝑊)
7 xpexg 6925 . . . . . 6 (({𝑗} ∈ V ∧ 𝐶𝑊) → ({𝑗} × 𝐶) ∈ V)
85, 6, 7sylancr 694 . . . . 5 ((𝜑𝑗𝐴) → ({𝑗} × 𝐶) ∈ V)
98ralrimiva 2962 . . . 4 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐶) ∈ V)
10 iunexg 7104 . . . 4 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐶) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐶) ∈ V)
114, 9, 10syl2anc 692 . . 3 (𝜑 𝑗𝐴 ({𝑗} × 𝐶) ∈ V)
12 gsum2d2.f . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)
1312ralrimivva 2967 . . . 4 (𝜑 → ∀𝑗𝐴𝑘𝐶 𝑋𝐵)
14 eqid 2621 . . . . 5 (𝑗𝐴, 𝑘𝐶𝑋) = (𝑗𝐴, 𝑘𝐶𝑋)
1514fmpt2x 7196 . . . 4 (∀𝑗𝐴𝑘𝐶 𝑋𝐵 ↔ (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
1613, 15sylib 208 . . 3 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
17 gsum2d2.u . . . 4 (𝜑𝑈 ∈ Fin)
18 gsum2d2.n . . . 4 ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
191, 2, 3, 4, 6, 12, 17, 18gsum2d2lem 18312 . . 3 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
20 relxp 5198 . . . . . . 7 Rel ({𝑘} × 𝐸)
2120rgenw 2920 . . . . . 6 𝑘𝐷 Rel ({𝑘} × 𝐸)
22 reliun 5210 . . . . . 6 (Rel 𝑘𝐷 ({𝑘} × 𝐸) ↔ ∀𝑘𝐷 Rel ({𝑘} × 𝐸))
2321, 22mpbir 221 . . . . 5 Rel 𝑘𝐷 ({𝑘} × 𝐸)
24 cnvf1o 7236 . . . . 5 (Rel 𝑘𝐷 ({𝑘} × 𝐸) → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑘𝐷 ({𝑘} × 𝐸))
2523, 24ax-mp 5 . . . 4 (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑘𝐷 ({𝑘} × 𝐸)
26 relxp 5198 . . . . . . . 8 Rel ({𝑗} × 𝐶)
2726rgenw 2920 . . . . . . 7 𝑗𝐴 Rel ({𝑗} × 𝐶)
28 reliun 5210 . . . . . . 7 (Rel 𝑗𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐶))
2927, 28mpbir 221 . . . . . 6 Rel 𝑗𝐴 ({𝑗} × 𝐶)
30 relcnv 5472 . . . . . 6 Rel 𝑘𝐷 ({𝑘} × 𝐸)
31 nfv 1840 . . . . . . . 8 𝑘𝜑
32 nfv 1840 . . . . . . . . 9 𝑘𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶)
33 nfiu1 4523 . . . . . . . . . . 11 𝑘 𝑘𝐷 ({𝑘} × 𝐸)
3433nfcnv 5271 . . . . . . . . . 10 𝑘 𝑘𝐷 ({𝑘} × 𝐸)
3534nfel2 2777 . . . . . . . . 9 𝑘𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)
3632, 35nfbi 1830 . . . . . . . 8 𝑘(⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))
3731, 36nfim 1822 . . . . . . 7 𝑘(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
38 opeq2 4378 . . . . . . . . . 10 (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
3938eleq1d 2683 . . . . . . . . 9 (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶)))
4038eleq1d 2683 . . . . . . . . 9 (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
4139, 40bibi12d 335 . . . . . . . 8 (𝑘 = 𝑦 → ((⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))))
4241imbi2d 330 . . . . . . 7 (𝑘 = 𝑦 → ((𝜑 → (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))) ↔ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))))
43 nfv 1840 . . . . . . . . 9 𝑗𝜑
44 nfiu1 4523 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐶)
4544nfel2 2777 . . . . . . . . . 10 𝑗𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶)
46 nfv 1840 . . . . . . . . . 10 𝑗𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)
4745, 46nfbi 1830 . . . . . . . . 9 𝑗(⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))
4843, 47nfim 1822 . . . . . . . 8 𝑗(𝜑 → (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
49 opeq1 4377 . . . . . . . . . . 11 (𝑗 = 𝑥 → ⟨𝑗, 𝑘⟩ = ⟨𝑥, 𝑘⟩)
5049eleq1d 2683 . . . . . . . . . 10 (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶)))
5149eleq1d 2683 . . . . . . . . . 10 (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
5250, 51bibi12d 335 . . . . . . . . 9 (𝑗 = 𝑥 → ((⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)) ↔ (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))))
5352imbi2d 330 . . . . . . . 8 (𝑗 = 𝑥 → ((𝜑 → (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))) ↔ (𝜑 → (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))))
54 gsumcom2.c . . . . . . . . . 10 (𝜑 → ((𝑗𝐴𝑘𝐶) ↔ (𝑘𝐷𝑗𝐸)))
55 opeliunxp 5141 . . . . . . . . . 10 (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑗𝐴𝑘𝐶))
56 opeliunxp 5141 . . . . . . . . . 10 (⟨𝑘, 𝑗⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸) ↔ (𝑘𝐷𝑗𝐸))
5754, 55, 563bitr4g 303 . . . . . . . . 9 (𝜑 → (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑘, 𝑗⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
58 vex 3193 . . . . . . . . . 10 𝑗 ∈ V
59 vex 3193 . . . . . . . . . 10 𝑘 ∈ V
6058, 59opelcnv 5274 . . . . . . . . 9 (⟨𝑗, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑘, 𝑗⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸))
6157, 60syl6bbr 278 . . . . . . . 8 (𝜑 → (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
6248, 53, 61chvar 2261 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
6337, 42, 62chvar 2261 . . . . . 6 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐷 ({𝑘} × 𝐸)))
6429, 30, 63eqrelrdv 5187 . . . . 5 (𝜑 𝑗𝐴 ({𝑗} × 𝐶) = 𝑘𝐷 ({𝑘} × 𝐸))
65 f1oeq3 6096 . . . . 5 ( 𝑗𝐴 ({𝑗} × 𝐶) = 𝑘𝐷 ({𝑘} × 𝐸) → ((𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑘𝐷 ({𝑘} × 𝐸)))
6664, 65syl 17 . . . 4 (𝜑 → ((𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑘𝐷 ({𝑘} × 𝐸)))
6725, 66mpbiri 248 . . 3 (𝜑 → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑗𝐴 ({𝑗} × 𝐶))
681, 2, 3, 11, 16, 19, 67gsumf1o 18257 . 2 (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg ((𝑗𝐴, 𝑘𝐶𝑋) ∘ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}))))
69 sneq 4165 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = {⟨𝑥, 𝑦⟩})
7069cnveqd 5268 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = {⟨𝑥, 𝑦⟩})
7170unieqd 4419 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = {⟨𝑥, 𝑦⟩})
72 opswap 5591 . . . . . . . . 9 {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥
7371, 72syl6eq 2671 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = ⟨𝑦, 𝑥⟩)
7473fveq2d 6162 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧}) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑦, 𝑥⟩))
75 df-ov 6618 . . . . . . 7 (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑦, 𝑥⟩)
7674, 75syl6eqr 2673 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧}) = (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
7776mpt2mptx 6716 . . . . 5 (𝑧 𝑥𝐷 ({𝑥} × 𝑥 / 𝑘𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})) = (𝑥𝐷, 𝑦𝑥 / 𝑘𝐸 ↦ (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
78 nfcv 2761 . . . . . . 7 𝑥({𝑘} × 𝐸)
79 nfcv 2761 . . . . . . . 8 𝑘{𝑥}
80 nfcsb1v 3535 . . . . . . . 8 𝑘𝑥 / 𝑘𝐸
8179, 80nfxp 5112 . . . . . . 7 𝑘({𝑥} × 𝑥 / 𝑘𝐸)
82 sneq 4165 . . . . . . . 8 (𝑘 = 𝑥 → {𝑘} = {𝑥})
83 csbeq1a 3528 . . . . . . . 8 (𝑘 = 𝑥𝐸 = 𝑥 / 𝑘𝐸)
8482, 83xpeq12d 5110 . . . . . . 7 (𝑘 = 𝑥 → ({𝑘} × 𝐸) = ({𝑥} × 𝑥 / 𝑘𝐸))
8578, 81, 84cbviun 4530 . . . . . 6 𝑘𝐷 ({𝑘} × 𝐸) = 𝑥𝐷 ({𝑥} × 𝑥 / 𝑘𝐸)
86 mpteq1 4707 . . . . . 6 ( 𝑘𝐷 ({𝑘} × 𝐸) = 𝑥𝐷 ({𝑥} × 𝑥 / 𝑘𝐸) → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})) = (𝑧 𝑥𝐷 ({𝑥} × 𝑥 / 𝑘𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})))
8785, 86ax-mp 5 . . . . 5 (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})) = (𝑧 𝑥𝐷 ({𝑥} × 𝑥 / 𝑘𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧}))
88 nfcv 2761 . . . . . 6 𝑥𝐸
89 nfcv 2761 . . . . . 6 𝑥(𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘)
90 nfcv 2761 . . . . . 6 𝑦(𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘)
91 nfcv 2761 . . . . . . 7 𝑘𝑦
92 nfmpt22 6688 . . . . . . 7 𝑘(𝑗𝐴, 𝑘𝐶𝑋)
93 nfcv 2761 . . . . . . 7 𝑘𝑥
9491, 92, 93nfov 6641 . . . . . 6 𝑘(𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥)
95 nfcv 2761 . . . . . . 7 𝑗𝑦
96 nfmpt21 6687 . . . . . . 7 𝑗(𝑗𝐴, 𝑘𝐶𝑋)
97 nfcv 2761 . . . . . . 7 𝑗𝑥
9895, 96, 97nfov 6641 . . . . . 6 𝑗(𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥)
99 oveq2 6623 . . . . . . 7 (𝑘 = 𝑥 → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
100 oveq1 6622 . . . . . . 7 (𝑗 = 𝑦 → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑥) = (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
10199, 100sylan9eq 2675 . . . . . 6 ((𝑘 = 𝑥𝑗 = 𝑦) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
10288, 80, 89, 90, 94, 98, 83, 101cbvmpt2x 6698 . . . . 5 (𝑘𝐷, 𝑗𝐸 ↦ (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘)) = (𝑥𝐷, 𝑦𝑥 / 𝑘𝐸 ↦ (𝑦(𝑗𝐴, 𝑘𝐶𝑋)𝑥))
10377, 87, 1023eqtr4i 2653 . . . 4 (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})) = (𝑘𝐷, 𝑗𝐸 ↦ (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘))
104 f1of 6104 . . . . . . 7 ((𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)–1-1-onto 𝑗𝐴 ({𝑗} × 𝐶) → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)⟶ 𝑗𝐴 ({𝑗} × 𝐶))
10567, 104syl 17 . . . . . 6 (𝜑 → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)⟶ 𝑗𝐴 ({𝑗} × 𝐶))
106 eqid 2621 . . . . . . 7 (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}) = (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧})
107106fmpt 6347 . . . . . 6 (∀𝑧 𝑘𝐷 ({𝑘} × 𝐸) {𝑧} ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}): 𝑘𝐷 ({𝑘} × 𝐸)⟶ 𝑗𝐴 ({𝑗} × 𝐶))
108105, 107sylibr 224 . . . . 5 (𝜑 → ∀𝑧 𝑘𝐷 ({𝑘} × 𝐸) {𝑧} ∈ 𝑗𝐴 ({𝑗} × 𝐶))
109 eqidd 2622 . . . . 5 (𝜑 → (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}) = (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}))
11016feqmptd 6216 . . . . 5 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) = (𝑥 𝑗𝐴 ({𝑗} × 𝐶) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑥)))
111 fveq2 6158 . . . . 5 (𝑥 = {𝑧} → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑥) = ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧}))
112108, 109, 110, 111fmptcof 6363 . . . 4 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) ∘ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧})) = (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ ((𝑗𝐴, 𝑘𝐶𝑋)‘ {𝑧})))
11312ex 450 . . . . . . . . 9 (𝜑 → ((𝑗𝐴𝑘𝐶) → 𝑋𝐵))
11414ovmpt4g 6748 . . . . . . . . . 10 ((𝑗𝐴𝑘𝐶𝑋𝐵) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
1151143expia 1264 . . . . . . . . 9 ((𝑗𝐴𝑘𝐶) → (𝑋𝐵 → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋))
116113, 115sylcom 30 . . . . . . . 8 (𝜑 → ((𝑗𝐴𝑘𝐶) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋))
11754, 116sylbird 250 . . . . . . 7 (𝜑 → ((𝑘𝐷𝑗𝐸) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋))
1181173impib 1259 . . . . . 6 ((𝜑𝑘𝐷𝑗𝐸) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
119118eqcomd 2627 . . . . 5 ((𝜑𝑘𝐷𝑗𝐸) → 𝑋 = (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘))
120119mpt2eq3dva 6684 . . . 4 (𝜑 → (𝑘𝐷, 𝑗𝐸𝑋) = (𝑘𝐷, 𝑗𝐸 ↦ (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘)))
121103, 112, 1203eqtr4a 2681 . . 3 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) ∘ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧})) = (𝑘𝐷, 𝑗𝐸𝑋))
122121oveq2d 6631 . 2 (𝜑 → (𝐺 Σg ((𝑗𝐴, 𝑘𝐶𝑋) ∘ (𝑧 𝑘𝐷 ({𝑘} × 𝐸) ↦ {𝑧}))) = (𝐺 Σg (𝑘𝐷, 𝑗𝐸𝑋)))
12368, 122eqtrd 2655 1 (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐷, 𝑗𝐸𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  csb 3519  {csn 4155  cop 4161   cuni 4409   ciun 4492   class class class wbr 4623  cmpt 4683   × cxp 5082  ccnv 5083  ccom 5088  Rel wrel 5089  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cmpt2 6617  Fincfn 7915  Basecbs 15800  0gc0g 16040   Σg cgsu 16041  CMndccmn 18133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-cntz 17690  df-cmn 18135
This theorem is referenced by:  gsumcom  18316  gsumbagdiag  19316
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