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Theorem gsumpropd2lem 17194
Description: Lemma for gsumpropd2 17195. (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 481 . . . . . 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 2623 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g𝐺)𝑡) = 𝑡 ↔ (𝑠(+g𝐻)𝑡) = 𝑡))
53oveqrspc2v 6627 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 6627 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
76ancom2s 843 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
87eqeq1d 2623 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g𝐺)𝑠) = 𝑡 ↔ (𝑡(+g𝐻)𝑠) = 𝑡))
94, 8anbi12d 746 . . . . . . 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 679 . . . . . 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 3143 . . . . 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 3182 . . . 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 3612 . . 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 2622 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
1514, 1, 3grpidpropd 17182 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
16 simprl 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
17 gsumpropd2.r . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1817ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19 gsumpropd2.n . . . . . . . . . . . . . 14 (𝜑 → Fun 𝐹)
2019ad2antrr 761 . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
2321, 22eleqtrrd 2701 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24 fvelrn 6308 . . . . . . . . . . . . 13 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
2520, 23, 24syl2anc 692 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
2618, 25sseldd 3584 . . . . . . . . . . 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 750 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
293adantlr 750 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
3016, 26, 28, 29seqfeq4 12790 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3130eqeq2d 2631 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3231anassrs 679 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3332pm5.32da 672 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3433rexbidva 3042 . . . . . 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 1847 . . . . 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 5831 . . . 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 3706 . . . . . . . . . . . . . . 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 5425 . . . . . . . . . . . . . 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 2680 . . . . . . . . . . . . 13 (𝜑𝐴 = 𝐵)
4241fveq2d 6152 . . . . . . . . . . . 12 (𝜑 → (#‘𝐴) = (#‘𝐵))
4342fveq2d 6152 . . . . . . . . . . 11 (𝜑 → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)))
4443adantr 481 . . . . . . . . . 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 765 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47 f1ofun 6096 . . . . . . . . . . . . . . . 16 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → Fun 𝑓)
4847ad3antlr 766 . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(#‘𝐴)))
5150ad3antlr 766 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐴)))
5242oveq2d 6620 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(#‘𝐴)) = (1...(#‘𝐵)))
5352ad3antrrr 765 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (1...(#‘𝐴)) = (1...(#‘𝐵)))
5451, 53eqtrd 2655 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐵)))
5549, 54eleqtrrd 2701 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ dom 𝑓)
56 fvco 6231 . . . . . . . . . . . . . . 15 ((Fun 𝑓𝑎 ∈ dom 𝑓) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5748, 55, 56syl2anc 692 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5819ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → Fun 𝐹)
59 difpreima 6299 . . . . . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
62 difss 3715 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62syl6eqss 3634 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5276 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 5554 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2643 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66syl6sseqr 3631 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6094 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))⟶𝐴)
7069ad3antlr 766 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑓:(1...(#‘𝐴))⟶𝐴)
7149, 53eleqtrrd 2701 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ (1...(#‘𝐴)))
7270, 71ffvelrnd 6316 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3584 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 6308 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ (𝑓𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7558, 73, 74syl2anc 692 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7657, 75eqeltrd 2698 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ ran 𝐹)
7746, 76sseldd 3584 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
78 simpll 789 . . . . . . . . . . . . 13 (((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) → 𝜑)
7927caovclg 6779 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8078, 79sylan 488 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ (#‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
8178, 5sylan 488 . . . . . . . . . . . 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 12790 . . . . . . . . . . 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 11351 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
85 seqfn 12753 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
86 fndm 5948 . . . . . . . . . . . . . . . . 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 2690 . . . . . . . . . . . . . . 15 ((#‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) ↔ (#‘𝐵) ∈ (ℤ‘1))
8983, 88sylnibr 319 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)))
90 ndmfv 6175 . . . . . . . . . . . . . 14 (¬ (#‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)) = ∅)
9189, 90syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)) = ∅)
92 seqfn 12753 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
93 fndm 5948 . . . . . . . . . . . . . . . . 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 2690 . . . . . . . . . . . . . . 15 ((#‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) ↔ (#‘𝐵) ∈ (ℤ‘1))
9683, 95sylnibr 319 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)))
97 ndmfv 6175 . . . . . . . . . . . . . 14 (¬ (#‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) → (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)) = ∅)
9896, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)) = ∅)
9991, 98eqtr4d 2658 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)))
10099adantlr 750 . . . . . . . . . . 11 (((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) ∧ ¬ (#‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)))
10182, 100pm2.61dan 831 . . . . . . . . . 10 ((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)))
10244, 101eqtrd 2655 . . . . . . . . 9 ((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐴)) = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)))
103102eqeq2d 2631 . . . . . . . 8 ((𝜑𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐴)) ↔ 𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵))))
104103pm5.32da 672 . . . . . . 7 (𝜑 → ((𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(#‘𝐴))) ↔ (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(#‘𝐵)))))
105 f1oeq2 6085 . . . . . . . . . 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 6086 . . . . . . . . . 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 740 . . . . . . 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 1847 . . . . 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 5831 . . . 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 4078 . . 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 4085 . 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 2621 . . 3 (Base‘𝐺) = (Base‘𝐺)
117 eqid 2621 . . 3 (0g𝐺) = (0g𝐺)
118 eqid 2621 . . 3 (+g𝐺) = (+g𝐺)
119 eqid 2621 . . 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 2622 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
124116, 117, 118, 119, 120, 121, 122, 123gsumvalx 17191 . 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 2621 . . 3 (Base‘𝐻) = (Base‘𝐻)
126 eqid 2621 . . 3 (0g𝐻) = (0g𝐻)
127 eqid 2621 . . 3 (+g𝐻) = (+g𝐻)
128 eqid 2621 . . 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 17191 . 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 2665 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  c0 3891  ifcif 4058  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  cio 5808  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  1c1 9881  cz 11321  cuz 11631  ...cfz 12268  seqcseq 12741  #chash 13057  Basecbs 15781  +gcplusg 15862  0gc0g 16021   Σg cgsu 16022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-seq 12742  df-0g 16023  df-gsum 16024
This theorem is referenced by:  gsumpropd2  17195
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