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Theorem gsumval3eu 19599
Description: The group sum as defined in gsumval3a 19598 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 2946 . . . . 5 (𝜑 → ¬ 𝑊 = ∅)
3 gsumval3a.t . . . . . . 7 (𝜑𝑊 ∈ Fin)
4 fz1f1o 15521 . . . . . . 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 862 . . . . 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 497 . . 3 (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
9 excom 2162 . . . 4 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
10 exancom 1864 . . . . . 6 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
11 fvex 6842 . . . . . . 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 3490 . . . . . 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 1850 . . . 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 233 . 2 (𝜑 → ∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
18 exdistrv 1959 . . . 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 654 . . . . . 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 482 . . . . . . . . . 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 18490 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25243expb 1120 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
2621, 25sylan 581 . . . . . . . . 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 482 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2928sselda 3935 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
3029adantrr 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31 simprr 771 . . . . . . . . . 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 19031 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3430, 31, 33syl2anc 585 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3522, 23mndass 18491 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
3621, 35sylan 581 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
377simpld 496 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℕ)
3837adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
39 nnuz 12726 . . . . . . . . . 10 ℕ = (ℤ‘1)
4038, 39eleqtrdi 2848 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ (ℤ‘1))
41 gsumval3.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
4241adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
4342frnd 6663 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹𝐵)
44 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)
45 f1ocnv 6783 . . . . . . . . . . 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 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
48 f1oco 6794 . . . . . . . . . 10 ((𝑔:𝑊1-1-onto→(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
4946, 47, 48syl2anc 585 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
50 f1of 6771 . . . . . . . . . . . 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 6927 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝑊𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5351, 52sylan 581 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5442ffnd 6656 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
55 gsumval3a.s . . . . . . . . . . . . . 14 (𝜑𝑊𝐴)
5655adantr 482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
5751, 56fssd 6673 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
5857ffvelcdmda 7021 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔𝑥) ∈ 𝐴)
59 fnfvelrn 7018 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑥) ∈ 𝐴) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6054, 58, 59syl2an2r 683 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6153, 60eqeltrd 2838 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) ∈ ran 𝐹)
62 f1of 6771 . . . . . . . . . . . . . . 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 6927 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6563, 64sylan 581 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6665fveq2d 6833 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((𝑔𝑓)‘𝑘)) = (𝑔‘(𝑔‘(𝑓𝑘))))
6763ffvelcdmda 7021 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) ∈ 𝑊)
68 f1ocnvfv2 7209 . . . . . . . . . . . . 13 ((𝑔:(1...(♯‘𝑊))–1-1-onto𝑊 ∧ (𝑓𝑘) ∈ 𝑊) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
6944, 67, 68syl2an2r 683 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7066, 69eqtr2d 2778 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) = (𝑔‘((𝑔𝑓)‘𝑘)))
7170fveq2d 6833 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
72 fvco3 6927 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
7363, 72sylan 581 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
74 f1of 6771 . . . . . . . . . . . . 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 7021 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
77 fvco3 6927 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝐴 ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
7857, 76, 77syl2an2r 683 . . . . . . . . . 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 2787 . . . . . . . . 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 13869 . . . . . . . 8 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
81 eqeq12 2754 . . . . . . . 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 455 . . . . . 6 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8419, 83syl5bir 243 . . . . 5 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8584exlimdvv 1937 . . . 4 (𝜑 → (∃𝑓𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8618, 85syl5bir 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 1931 . 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 2741 . . . . . 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 1924 . . . 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 6759 . . . . . 6 (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊))
92 coeq2 5804 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
9392seqeq3d 13834 . . . . . . . 8 (𝑓 = 𝑔 → seq1( + , (𝐹𝑓)) = seq1( + , (𝐹𝑔)))
9493fveq1d 6831 . . . . . . 7 (𝑓 = 𝑔 → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
9594eqeq2d 2748 . . . . . 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 2040 . . . 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 2616 . 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 584 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 397  wo 845  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2567  wne 2941  wss 3901  c0 4273  ccnv 5623  ran crn 5625  ccom 5628   Fn wfn 6478  wf 6479  1-1-ontowf1o 6482  cfv 6483  (class class class)co 7341  Fincfn 8808  1c1 10977  cn 12078  cuz 12687  ...cfz 13344  seqcseq 13826  chash 14149  Basecbs 17009  +gcplusg 17059  0gc0g 17247  Mndcmnd 18482  Cntzccntz 19017
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-er 8573  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-card 9800  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-n0 12339  df-z 12425  df-uz 12688  df-fz 13345  df-fzo 13488  df-seq 13827  df-hash 14150  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-cntz 19019
This theorem is referenced by:  gsumval3lem2  19601
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