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Theorem gsumpropd2lem 18642
Description: Lemma for gsumpropd2 18643. (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 479 . . . . . 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 2727 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g𝐺)𝑡) = 𝑡 ↔ (𝑠(+g𝐻)𝑡) = 𝑡))
53oveqrspc2v 7446 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 7446 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
76ancom2s 648 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
87eqeq1d 2727 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g𝐺)𝑠) = 𝑡 ↔ (𝑡(+g𝐻)𝑠) = 𝑡))
94, 8anbi12d 630 . . . . . . 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 466 . . . . . 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 3316 . . . . 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 3435 . . . 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 4009 . . 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 2726 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
1514, 1, 3grpidpropd 18625 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
16 simprl 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
17 gsumpropd2.r . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1817ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19 gsumpropd2.n . . . . . . . . . . . . . 14 (𝜑 → Fun 𝐹)
2019ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21 simpr 483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22 simplrr 776 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
2321, 22eleqtrrd 2828 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24 fvelrn 7085 . . . . . . . . . . . . 13 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
2520, 23, 24syl2anc 582 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
2618, 25sseldd 3977 . . . . . . . . . . 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 713 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
293adantlr 713 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
3016, 26, 28, 29seqfeq4 14052 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3130eqeq2d 2736 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3231anassrs 466 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3332pm5.32da 577 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3433rexbidva 3166 . . . . . 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 1916 . . . . 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 6533 . . . 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 4118 . . . . . . . . . . . . . . 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 6064 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 13 (𝜑𝐴 = 𝐵)
4241fveq2d 6900 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) = (♯‘𝐵))
4342fveq2d 6900 . . . . . . . . . . 11 (𝜑 → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
4443adantr 479 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
45 simpr 483 . . . . . . . . . . . 12 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (♯‘𝐵) ∈ (ℤ‘1))
4617ad3antrrr 728 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47 f1ofun 6840 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑓)
4847ad3antlr 729 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
49 simpr 483 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐵)))
50 f1odm 6842 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
5150ad3antlr 729 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐴)))
5242oveq2d 7435 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5352ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5451, 53eqtrd 2765 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
5549, 54eleqtrrd 2828 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ dom 𝑓)
56 fvco 6995 . . . . . . . . . . . . . . 15 ((Fun 𝑓𝑎 ∈ dom 𝑓) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5748, 55, 56syl2anc 582 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5819ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝐹)
59 difpreima 7073 . . . . . . . . . . . . . . . . . . . . 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, 60eqtrid 2777 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
62 difss 4128 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62eqsstrdi 4031 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5898 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 6208 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2753 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66sseqtrrdi 4028 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6838 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
7069ad3antlr 729 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
7149, 53eleqtrrd 2828 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐴)))
7270, 71ffvelcdmd 7094 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3977 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 7085 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ (𝑓𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7558, 73, 74syl2anc 582 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7657, 75eqeltrd 2825 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ ran 𝐹)
7746, 76sseldd 3977 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
7827caovclg 7613 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
7978ad4ant14 750 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
805ad4ant14 750 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
8145, 77, 79, 80seqfeq4 14052 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
82 simpr 483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ (ℤ‘1))
83 1z 12625 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
84 seqfn 14014 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
85 fndm 6658 . . . . . . . . . . . . . . . . 17 (seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1) → dom seq1((+g𝐺), (𝐹𝑓)) = (ℤ‘1))
8683, 84, 85mp2b 10 . . . . . . . . . . . . . . . 16 dom seq1((+g𝐺), (𝐹𝑓)) = (ℤ‘1)
8786eleq2i 2817 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
8882, 87sylnibr 328 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)))
89 ndmfv 6931 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9088, 89syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
91 seqfn 14014 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
92 fndm 6658 . . . . . . . . . . . . . . . . 17 (seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1) → dom seq1((+g𝐻), (𝐹𝑓)) = (ℤ‘1))
9383, 91, 92mp2b 10 . . . . . . . . . . . . . . . 16 dom seq1((+g𝐻), (𝐹𝑓)) = (ℤ‘1)
9493eleq2i 2817 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
9582, 94sylnibr 328 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)))
96 ndmfv 6931 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9795, 96syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9890, 97eqtr4d 2768 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
9998adantlr 713 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10081, 99pm2.61dan 811 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10144, 100eqtrd 2765 . . . . . . . . 9 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
102101eqeq2d 2736 . . . . . . . 8 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) ↔ 𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))))
103102pm5.32da 577 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
10452f1oeq2d 6834 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐴))
10541f1oeq3d 6835 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
106104, 105bitrd 278 . . . . . . . 8 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
107106anbi1d 629 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
108103, 107bitrd 278 . . . . . 6 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
109108exbidv 1916 . . . . 5 (𝜑 → (∃𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ ∃𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
110109iotabidv 6533 . . . 4 (𝜑 → (℩𝑥𝑓(𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)))) = (℩𝑥𝑓(𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
11136, 110ifeq12d 4551 . . 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))))))
11213, 15, 111ifbieq12d 4558 . 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)))))))
113 eqid 2725 . . 3 (Base‘𝐺) = (Base‘𝐺)
114 eqid 2725 . . 3 (0g𝐺) = (0g𝐺)
115 eqid 2725 . . 3 (+g𝐺) = (+g𝐺)
116 eqid 2725 . . 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)}
11739a1i 11 . . 3 (𝜑𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
118 gsumpropd2.g . . 3 (𝜑𝐺𝑊)
119 gsumpropd2.f . . 3 (𝜑𝐹𝑉)
120 eqidd 2726 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
121113, 114, 115, 116, 117, 118, 119, 120gsumvalx 18639 . 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)))))))
122 eqid 2725 . . 3 (Base‘𝐻) = (Base‘𝐻)
123 eqid 2725 . . 3 (0g𝐻) = (0g𝐻)
124 eqid 2725 . . 3 (+g𝐻) = (+g𝐻)
125 eqid 2725 . . 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)}
12640a1i 11 . . 3 (𝜑𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)})))
127 gsumpropd2.h . . 3 (𝜑𝐻𝑋)
128122, 123, 124, 125, 126, 127, 119, 120gsumvalx 18639 . 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)))))))
129112, 121, 1283eqtr4d 2775 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  wss 3944  c0 4322  ifcif 4530  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  ccom 5682  cio 6499  Fun wfun 6543   Fn wfn 6544  wf 6545  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  1c1 11141  cz 12591  cuz 12855  ...cfz 13519  seqcseq 14002  chash 14325  Basecbs 17183  +gcplusg 17236  0gc0g 17424   Σg cgsu 17425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-seq 14003  df-0g 17426  df-gsum 17427
This theorem is referenced by:  gsumpropd2  18643
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