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Theorem gsumpropd2lem 17470
Description: Lemma for gsumpropd2 17471. (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 2758 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g𝐺)𝑡) = 𝑡 ↔ (𝑠(+g𝐻)𝑡) = 𝑡))
53oveqrspc2v 6832 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 6832 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
76ancom2s 879 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
87eqeq1d 2758 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g𝐺)𝑠) = 𝑡 ↔ (𝑡(+g𝐻)𝑠) = 𝑡))
94, 8anbi12d 749 . . . . . . 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 683 . . . . . 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 3289 . . . . 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 3332 . . . 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 3770 . . 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 2757 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
1514, 1, 3grpidpropd 17458 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
16 simprl 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
17 gsumpropd2.r . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1817ad2antrr 764 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19 gsumpropd2.n . . . . . . . . . . . . . 14 (𝜑 → Fun 𝐹)
2019ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21 simpr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22 simplrr 820 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
2321, 22eleqtrrd 2838 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24 fvelrn 6511 . . . . . . . . . . . . 13 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
2520, 23, 24syl2anc 696 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
2618, 25sseldd 3741 . . . . . . . . . . 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 753 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
293adantlr 753 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
3016, 26, 28, 29seqfeq4 13040 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3130eqeq2d 2766 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3231anassrs 683 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3332pm5.32da 676 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3433rexbidva 3183 . . . . . 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 1995 . . . . 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 6029 . . . 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 3867 . . . . . . . . . . . . . . 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 5620 . . . . . . . . . . . . . 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 2815 . . . . . . . . . . . . 13 (𝜑𝐴 = 𝐵)
4241fveq2d 6352 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) = (♯‘𝐵))
4342fveq2d 6352 . . . . . . . . . . 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 479 . . . . . . . . . . . 12 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (♯‘𝐵) ∈ (ℤ‘1))
4617ad3antrrr 768 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47 f1ofun 6296 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑓)
4847ad3antlr 769 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
49 simpr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐵)))
50 f1odm 6298 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
5150ad3antlr 769 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐴)))
5242oveq2d 6825 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5352ad3antrrr 768 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5451, 53eqtrd 2790 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
5549, 54eleqtrrd 2838 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ dom 𝑓)
56 fvco 6432 . . . . . . . . . . . . . . 15 ((Fun 𝑓𝑎 ∈ dom 𝑓) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5748, 55, 56syl2anc 696 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5819ad3antrrr 768 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝐹)
59 difpreima 6502 . . . . . . . . . . . . . . . . . . . . 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 2802 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
62 difss 3876 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62syl6eqss 3792 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5467 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 5749 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2778 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66syl6sseqr 3789 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 768 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6294 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
7069ad3antlr 769 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
7149, 53eleqtrrd 2838 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐴)))
7270, 71ffvelrnd 6519 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3741 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 6511 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ (𝑓𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7558, 73, 74syl2anc 696 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7657, 75eqeltrd 2835 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ ran 𝐹)
7746, 76sseldd 3741 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
78 simpll 807 . . . . . . . . . . . . 13 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → 𝜑)
7927caovclg 6987 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8078, 79sylan 489 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8178, 5sylan 489 . . . . . . . . . . . 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 13040 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
83 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ (ℤ‘1))
84 1z 11595 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
85 seqfn 13003 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
86 fndm 6147 . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
8983, 88sylnibr 318 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)))
90 ndmfv 6375 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9189, 90syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
92 seqfn 13003 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
93 fndm 6147 . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
9683, 95sylnibr 318 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)))
97 ndmfv 6375 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9896, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9991, 98eqtr4d 2793 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10099adantlr 753 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10182, 100pm2.61dan 867 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10244, 101eqtrd 2790 . . . . . . . . 9 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
103102eqeq2d 2766 . . . . . . . 8 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) ↔ 𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))))
104103pm5.32da 676 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
105 f1oeq2 6285 . . . . . . . . . 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 6286 . . . . . . . . . 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 268 . . . . . . . 8 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
110109anbi1d 743 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
111104, 110bitrd 268 . . . . . 6 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
112111exbidv 1995 . . . . 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 6029 . . . 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 4246 . . 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 4253 . 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 2756 . . 3 (Base‘𝐺) = (Base‘𝐺)
117 eqid 2756 . . 3 (0g𝐺) = (0g𝐺)
118 eqid 2756 . . 3 (+g𝐺) = (+g𝐺)
119 eqid 2756 . . 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 2757 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
124116, 117, 118, 119, 120, 121, 122, 123gsumvalx 17467 . 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 2756 . . 3 (Base‘𝐻) = (Base‘𝐻)
126 eqid 2756 . . 3 (0g𝐻) = (0g𝐻)
127 eqid 2756 . . 3 (+g𝐻) = (+g𝐻)
128 eqid 2756 . . 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 17467 . 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 2800 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1628  wex 1849  wcel 2135  wral 3046  wrex 3047  {crab 3050  Vcvv 3336  cdif 3708  wss 3711  c0 4054  ifcif 4226  ccnv 5261  dom cdm 5262  ran crn 5263  cima 5265  ccom 5266  cio 6006  Fun wfun 6039   Fn wfn 6040  wf 6041  1-1-ontowf1o 6044  cfv 6045  (class class class)co 6809  1c1 10125  cz 11565  cuz 11875  ...cfz 12515  seqcseq 12991  chash 13307  Basecbs 16055  +gcplusg 16139  0gc0g 16298   Σg cgsu 16299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-er 7907  df-en 8118  df-dom 8119  df-sdom 8120  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-n0 11481  df-z 11566  df-uz 11876  df-fz 12516  df-seq 12992  df-0g 16300  df-gsum 16301
This theorem is referenced by:  gsumpropd2  17471
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