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Theorem gsumpropd2lem 17541
Description: Lemma for gsumpropd2 17542. (Contributed by Thierry Arnoux, 28-Jun-2017.)
Hypotheses
Ref Expression
gsumpropd2.f (𝜑𝐹𝑉)
gsumpropd2.g (𝜑𝐺𝑊)
gsumpropd2.h (𝜑𝐻𝑋)
gsumpropd2.b (𝜑 → (Base‘𝐺) = (Base‘𝐻))
gsumpropd2.c ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
gsumpropd2.e ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
gsumpropd2.n (𝜑 → Fun 𝐹)
gsumpropd2.r (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
gsumprop2dlem.1 𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}))
gsumprop2dlem.2 𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))
Assertion
Ref Expression
gsumpropd2lem (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Distinct variable groups:   𝑡,𝑠,𝐹   𝐺,𝑠,𝑡   𝐻,𝑠,𝑡   𝜑,𝑠,𝑡
Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑡,𝑠)   𝑉(𝑡,𝑠)   𝑊(𝑡,𝑠)   𝑋(𝑡,𝑠)

Proof of Theorem gsumpropd2lem
Dummy variables 𝑎 𝑏 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumpropd2.b . . . . 5 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
21adantr 472 . . . . . 6 ((𝜑𝑠 ∈ (Base‘𝐺)) → (Base‘𝐺) = (Base‘𝐻))
3 gsumpropd2.e . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
43eqeq1d 2767 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g𝐺)𝑡) = 𝑡 ↔ (𝑠(+g𝐻)𝑡) = 𝑡))
53oveqrspc2v 6869 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 6869 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
76ancom2s 640 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
87eqeq1d 2767 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g𝐺)𝑠) = 𝑡 ↔ (𝑡(+g𝐻)𝑠) = 𝑡))
94, 8anbi12d 624 . . . . . . 7 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡) ↔ ((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)))
109anassrs 459 . . . . . 6 (((𝜑𝑠 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ (Base‘𝐺)) → (((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡) ↔ ((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)))
112, 10raleqbidva 3302 . . . . 5 ((𝜑𝑠 ∈ (Base‘𝐺)) → (∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡) ↔ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)))
121, 11rabeqbidva 3345 . . . 4 (𝜑 → {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)})
1312sseq2d 3793 . . 3 (𝜑 → (ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)} ↔ ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))
14 eqidd 2766 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
1514, 1, 3grpidpropd 17529 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
16 simprl 787 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
17 gsumpropd2.r . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1817ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19 gsumpropd2.n . . . . . . . . . . . . . 14 (𝜑 → Fun 𝐹)
2019ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21 simpr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22 simplrr 796 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
2321, 22eleqtrrd 2847 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24 fvelrn 6542 . . . . . . . . . . . . 13 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
2520, 23, 24syl2anc 579 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
2618, 25sseldd 3762 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ (Base‘𝐺))
27 gsumpropd2.c . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
2827adantlr 706 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
293adantlr 706 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
3016, 26, 28, 29seqfeq4 13057 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3130eqeq2d 2775 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3231anassrs 459 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3332pm5.32da 574 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3433rexbidva 3196 . . . . . 6 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3534exbidv 2016 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3635iotabidv 6052 . . . 4 (𝜑 → (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛))) = (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3712difeq2d 3890 . . . . . . . . . . . . . . 15 (𝜑 → (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}) = (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))
3837imaeq2d 5648 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)})))
39 gsumprop2dlem.1 . . . . . . . . . . . . . 14 𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}))
40 gsumprop2dlem.2 . . . . . . . . . . . . . 14 𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))
4138, 39, 403eqtr4g 2824 . . . . . . . . . . . . 13 (𝜑𝐴 = 𝐵)
4241fveq2d 6379 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) = (♯‘𝐵))
4342fveq2d 6379 . . . . . . . . . . 11 (𝜑 → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
4443adantr 472 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
45 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (♯‘𝐵) ∈ (ℤ‘1))
4617ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47 f1ofun 6322 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑓)
4847ad3antlr 722 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
49 simpr 477 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐵)))
50 f1odm 6324 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
5150ad3antlr 722 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐴)))
5242oveq2d 6858 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5352ad3antrrr 721 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5451, 53eqtrd 2799 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
5549, 54eleqtrrd 2847 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ dom 𝑓)
56 fvco 6463 . . . . . . . . . . . . . . 15 ((Fun 𝑓𝑎 ∈ dom 𝑓) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5748, 55, 56syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5819ad3antrrr 721 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝐹)
59 difpreima 6533 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝐹 → (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
6019, 59syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
6139, 60syl5eq 2811 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
62 difss 3899 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62syl6eqss 3815 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5484 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 5778 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2787 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66syl6sseqr 3812 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 721 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6320 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
7069ad3antlr 722 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
7149, 53eleqtrrd 2847 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐴)))
7270, 71ffvelrnd 6550 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3762 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 6542 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ (𝑓𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7558, 73, 74syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7657, 75eqeltrd 2844 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ ran 𝐹)
7746, 76sseldd 3762 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
78 simpll 783 . . . . . . . . . . . . 13 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → 𝜑)
7927caovclg 7024 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8078, 79sylan 575 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8178, 5sylan 575 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
8245, 77, 80, 81seqfeq4 13057 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
83 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ (ℤ‘1))
84 1z 11654 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
85 seqfn 13020 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
86 fndm 6168 . . . . . . . . . . . . . . . . 17 (seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1) → dom seq1((+g𝐺), (𝐹𝑓)) = (ℤ‘1))
8784, 85, 86mp2b 10 . . . . . . . . . . . . . . . 16 dom seq1((+g𝐺), (𝐹𝑓)) = (ℤ‘1)
8887eleq2i 2836 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
8983, 88sylnibr 320 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)))
90 ndmfv 6405 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9189, 90syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
92 seqfn 13020 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
93 fndm 6168 . . . . . . . . . . . . . . . . 17 (seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1) → dom seq1((+g𝐻), (𝐹𝑓)) = (ℤ‘1))
9484, 92, 93mp2b 10 . . . . . . . . . . . . . . . 16 dom seq1((+g𝐻), (𝐹𝑓)) = (ℤ‘1)
9594eleq2i 2836 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
9683, 95sylnibr 320 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)))
97 ndmfv 6405 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9896, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9991, 98eqtr4d 2802 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10099adantlr 706 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10182, 100pm2.61dan 847 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10244, 101eqtrd 2799 . . . . . . . . 9 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
103102eqeq2d 2775 . . . . . . . 8 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) ↔ 𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))))
104103pm5.32da 574 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
105 f1oeq2 6311 . . . . . . . . . 10 ((1...(♯‘𝐴)) = (1...(♯‘𝐵)) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐴))
10652, 105syl 17 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐴))
107 f1oeq3 6312 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
10841, 107syl 17 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
109106, 108bitrd 270 . . . . . . . 8 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
110109anbi1d 623 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
111104, 110bitrd 270 . . . . . 6 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
112111exbidv 2016 . . . . 5 (𝜑 → (∃𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ ∃𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
113112iotabidv 6052 . . . 4 (𝜑 → (℩𝑥𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)))) = (℩𝑥𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
11436, 113ifeq12d 4263 . . 3 (𝜑 → if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))))) = if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))))))
11513, 15, 114ifbieq12d 4270 . 2 (𝜑 → if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}, (0g𝐺), if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)))))) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}, (0g𝐻), if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))))
116 eqid 2765 . . 3 (Base‘𝐺) = (Base‘𝐺)
117 eqid 2765 . . 3 (0g𝐺) = (0g𝐺)
118 eqid 2765 . . 3 (+g𝐺) = (+g𝐺)
119 eqid 2765 . . 3 {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}
12039a1i 11 . . 3 (𝜑𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
121 gsumpropd2.g . . 3 (𝜑𝐺𝑊)
122 gsumpropd2.f . . 3 (𝜑𝐹𝑉)
123 eqidd 2766 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
124116, 117, 118, 119, 120, 121, 122, 123gsumvalx 17538 . 2 (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}, (0g𝐺), if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)))))))
125 eqid 2765 . . 3 (Base‘𝐻) = (Base‘𝐻)
126 eqid 2765 . . 3 (0g𝐻) = (0g𝐻)
127 eqid 2765 . . 3 (+g𝐻) = (+g𝐻)
128 eqid 2765 . . 3 {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}
12940a1i 11 . . 3 (𝜑𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)})))
130 gsumpropd2.h . . 3 (𝜑𝐻𝑋)
131125, 126, 127, 128, 129, 130, 122, 123gsumvalx 17538 . 2 (𝜑 → (𝐻 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}, (0g𝐻), if(dom 𝐹 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))))
132115, 124, 1313eqtr4d 2809 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  wss 3732  c0 4079  ifcif 4243  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  ccom 5281  cio 6029  Fun wfun 6062   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  1c1 10190  cz 11624  cuz 11886  ...cfz 12533  seqcseq 13008  chash 13321  Basecbs 16132  +gcplusg 16216  0gc0g 16368   Σg cgsu 16369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-seq 13009  df-0g 16370  df-gsum 16371
This theorem is referenced by:  gsumpropd2  17542
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