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Theorem gsumval3eu 19923
Description: The group sum as defined in gsumval3a 19922 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
gsumval3.b 𝐵 = (Base‘𝐺)
gsumval3.0 0 = (0g𝐺)
gsumval3.p + = (+g𝐺)
gsumval3.z 𝑍 = (Cntz‘𝐺)
gsumval3.g (𝜑𝐺 ∈ Mnd)
gsumval3.a (𝜑𝐴𝑉)
gsumval3.f (𝜑𝐹:𝐴𝐵)
gsumval3.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3a.t (𝜑𝑊 ∈ Fin)
gsumval3a.n (𝜑𝑊 ≠ ∅)
gsumval3a.s (𝜑𝑊𝐴)
Assertion
Ref Expression
gsumval3eu (𝜑 → ∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
Distinct variable groups:   𝑥,𝑓, +   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥, 0   𝑓,𝐺,𝑥   𝑥,𝑉   𝐵,𝑓,𝑥   𝑓,𝐹,𝑥   𝑓,𝑊,𝑥
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem gsumval3eu
Dummy variables 𝑔 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3a.n . . . . . 6 (𝜑𝑊 ≠ ∅)
21neneqd 2944 . . . . 5 (𝜑 → ¬ 𝑊 = ∅)
3 gsumval3a.t . . . . . . 7 (𝜑𝑊 ∈ Fin)
4 fz1f1o 15747 . . . . . . 7 (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
53, 4syl 17 . . . . . 6 (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
65ord 864 . . . . 5 (𝜑 → (¬ 𝑊 = ∅ → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
72, 6mpd 15 . . . 4 (𝜑 → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
87simprd 495 . . 3 (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
9 excom 2161 . . . 4 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
10 exancom 1860 . . . . . 6 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
11 fvex 6918 . . . . . . 7 (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∈ V
12 biidd 262 . . . . . . 7 (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
1311, 12ceqsexv 3531 . . . . . 6 (∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1410, 13bitri 275 . . . . 5 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1514exbii 1847 . . . 4 (∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
169, 15bitri 275 . . 3 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
178, 16sylibr 234 . 2 (𝜑 → ∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
18 exdistrv 1954 . . . 4 (∃𝑓𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) ↔ (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
19 an4 656 . . . . . 6 (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) ↔ ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
20 gsumval3.g . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
2120adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐺 ∈ Mnd)
22 gsumval3.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
23 gsumval3.p . . . . . . . . . . . 12 + = (+g𝐺)
2422, 23mndcl 18756 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25243expb 1120 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
2621, 25sylan 580 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
27 gsumval3.c . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2827adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2928sselda 3982 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
3029adantrr 717 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31 simprr 772 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑦 ∈ ran 𝐹)
32 gsumval3.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
3323, 32cntzi 19348 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3430, 31, 33syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3522, 23mndass 18757 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
3621, 35sylan 580 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
377simpld 494 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℕ)
3837adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
39 nnuz 12922 . . . . . . . . . 10 ℕ = (ℤ‘1)
4038, 39eleqtrdi 2850 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ (ℤ‘1))
41 gsumval3.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
4241adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
4342frnd 6743 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹𝐵)
44 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)
45 f1ocnv 6859 . . . . . . . . . . 11 (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:𝑊1-1-onto→(1...(♯‘𝑊)))
4644, 45syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:𝑊1-1-onto→(1...(♯‘𝑊)))
47 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
48 f1oco 6870 . . . . . . . . . 10 ((𝑔:𝑊1-1-onto→(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
4946, 47, 48syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
50 f1of 6847 . . . . . . . . . . . 12 (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))⟶𝑊)
5144, 50syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝑊)
52 fvco3 7007 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝑊𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5351, 52sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5442ffnd 6736 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
55 gsumval3a.s . . . . . . . . . . . . . 14 (𝜑𝑊𝐴)
5655adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
5751, 56fssd 6752 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
5857ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔𝑥) ∈ 𝐴)
59 fnfvelrn 7099 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑥) ∈ 𝐴) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6054, 58, 59syl2an2r 685 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6153, 60eqeltrd 2840 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) ∈ ran 𝐹)
62 f1of 6847 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))⟶𝑊)
6347, 62syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
64 fvco3 7007 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6563, 64sylan 580 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6665fveq2d 6909 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((𝑔𝑓)‘𝑘)) = (𝑔‘(𝑔‘(𝑓𝑘))))
6763ffvelcdmda 7103 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) ∈ 𝑊)
68 f1ocnvfv2 7298 . . . . . . . . . . . . 13 ((𝑔:(1...(♯‘𝑊))–1-1-onto𝑊 ∧ (𝑓𝑘) ∈ 𝑊) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
6944, 67, 68syl2an2r 685 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7066, 69eqtr2d 2777 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) = (𝑔‘((𝑔𝑓)‘𝑘)))
7170fveq2d 6909 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
72 fvco3 7007 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
7363, 72sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
74 f1of 6847 . . . . . . . . . . . . 13 ((𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7549, 74syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7675ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
77 fvco3 7007 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝐴 ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
7857, 76, 77syl2an2r 685 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
7971, 73, 783eqtr4d 2786 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)))
8026, 34, 36, 40, 43, 49, 61, 79seqf1o 14085 . . . . . . . 8 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
81 eqeq12 2753 . . . . . . . 8 ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
8280, 81syl5ibrcom 247 . . . . . . 7 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → 𝑥 = 𝑦))
8382expimpd 453 . . . . . 6 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8419, 83biimtrrid 243 . . . . 5 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8584exlimdvv 1933 . . . 4 (𝜑 → (∃𝑓𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8618, 85biimtrrid 243 . . 3 (𝜑 → ((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8786alrimivv 1927 . 2 (𝜑 → ∀𝑥𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
88 eqeq1 2740 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
8988anbi2d 630 . . . . 5 (𝑥 = 𝑦 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
9089exbidv 1920 . . . 4 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
91 f1oeq1 6835 . . . . . 6 (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊))
92 coeq2 5868 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
9392seqeq3d 14051 . . . . . . . 8 (𝑓 = 𝑔 → seq1( + , (𝐹𝑓)) = seq1( + , (𝐹𝑔)))
9493fveq1d 6907 . . . . . . 7 (𝑓 = 𝑔 → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
9594eqeq2d 2747 . . . . . 6 (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
9691, 95anbi12d 632 . . . . 5 (𝑓 = 𝑔 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
9796cbvexvw 2035 . . . 4 (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
9890, 97bitrdi 287 . . 3 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
9998eu4 2614 . 2 (∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∀𝑥𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦)))
10017, 87, 99sylanbrc 583 1 (𝜑 → ∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086  wal 1537   = wceq 1539  wex 1778  wcel 2107  ∃!weu 2567  wne 2939  wss 3950  c0 4332  ccnv 5683  ran crn 5685  ccom 5688   Fn wfn 6555  wf 6556  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  Fincfn 8986  1c1 11157  cn 12267  cuz 12879  ...cfz 13548  seqcseq 14043  chash 14370  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Mndcmnd 18748  Cntzccntz 19334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-cntz 19336
This theorem is referenced by:  gsumval3lem2  19925
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