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Theorem gsumpropd2lem 18542
Description: Lemma for gsumpropd2 18543. (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 482 . . . . . 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 2735 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g𝐺)𝑡) = 𝑡 ↔ (𝑠(+g𝐻)𝑡) = 𝑡))
53oveqrspc2v 7388 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 7388 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
76ancom2s 649 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g𝐺)𝑠) = (𝑡(+g𝐻)𝑠))
87eqeq1d 2735 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g𝐺)𝑠) = 𝑡 ↔ (𝑡(+g𝐻)𝑠) = 𝑡))
94, 8anbi12d 632 . . . . . . 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 469 . . . . . 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 3320 . . . . 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 3422 . . . 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 3980 . . 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 2734 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
1514, 1, 3grpidpropd 18525 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
16 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
17 gsumpropd2.r . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1817ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19 gsumpropd2.n . . . . . . . . . . . . . 14 (𝜑 → Fun 𝐹)
2019ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21 simpr 486 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22 simplrr 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
2321, 22eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24 fvelrn 7031 . . . . . . . . . . . . 13 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
2520, 23, 24syl2anc 585 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
2618, 25sseldd 3949 . . . . . . . . . . 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 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
293adantlr 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
3016, 26, 28, 29seqfeq4 13966 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3130eqeq2d 2744 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3231anassrs 469 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3332pm5.32da 580 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3433rexbidva 3170 . . . . . 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 1925 . . . . 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 6484 . . . 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 4086 . . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . 13 (𝜑𝐴 = 𝐵)
4241fveq2d 6850 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) = (♯‘𝐵))
4342fveq2d 6850 . . . . . . . . . . 11 (𝜑 → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
4443adantr 482 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)))
45 simpr 486 . . . . . . . . . . . 12 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (♯‘𝐵) ∈ (ℤ‘1))
4617ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47 f1ofun 6790 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑓)
4847ad3antlr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
49 simpr 486 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐵)))
50 f1odm 6792 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
5150ad3antlr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐴)))
5242oveq2d 7377 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5352ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
5451, 53eqtrd 2773 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
5549, 54eleqtrrd 2837 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ dom 𝑓)
56 fvco 6943 . . . . . . . . . . . . . . 15 ((Fun 𝑓𝑎 ∈ dom 𝑓) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5748, 55, 56syl2anc 585 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) = (𝐹‘(𝑓𝑎)))
5819ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝐹)
59 difpreima 7019 . . . . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})))
62 difss 4095 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62eqsstrdi 4002 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5855 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 6158 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2761 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66sseqtrrdi 3999 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6788 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
7069ad3antlr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
7149, 53eleqtrrd 2837 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐴)))
7270, 71ffvelcdmd 7040 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3949 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 7031 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ (𝑓𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7558, 73, 74syl2anc 585 . . . . . . . . . . . . . 14 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝐹‘(𝑓𝑎)) ∈ ran 𝐹)
7657, 75eqeltrd 2834 . . . . . . . . . . . . 13 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ ran 𝐹)
7746, 76sseldd 3949 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
7827caovclg 7550 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
7978ad4ant14 751 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) ∈ (Base‘𝐺))
805ad4ant14 751 . . . . . . . . . . . 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 13966 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
82 simpr 486 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ (ℤ‘1))
83 1z 12541 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
84 seqfn 13927 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
85 fndm 6609 . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
8882, 87sylnibr 329 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)))
89 ndmfv 6881 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9088, 89syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
91 seqfn 13927 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
92 fndm 6609 . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 15 ((♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) ↔ (♯‘𝐵) ∈ (ℤ‘1))
9582, 94sylnibr 329 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)))
96 ndmfv 6881 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐻), (𝐹𝑓)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9795, 96syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9890, 97eqtr4d 2776 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
9998adantlr 714 . . . . . . . . . . 11 (((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10081, 99pm2.61dan 812 . . . . . . . . . 10 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
10144, 100eqtrd 2773 . . . . . . . . 9 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))
102101eqeq2d 2744 . . . . . . . 8 ((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴)) ↔ 𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))))
103102pm5.32da 580 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
10452f1oeq2d 6784 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐴))
10541f1oeq3d 6785 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
106104, 105bitrd 279 . . . . . . . 8 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐵))
107106anbi1d 631 . . . . . . 7 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
108103, 107bitrd 279 . . . . . 6 (𝜑 → ((𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑥 = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑥 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘𝐵)))))
109108exbidv 1925 . . . . 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 6484 . . . 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 4511 . . 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 4518 . 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 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
114 eqid 2733 . . 3 (0g𝐺) = (0g𝐺)
115 eqid 2733 . . 3 (+g𝐺) = (+g𝐺)
116 eqid 2733 . . 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 2734 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
121113, 114, 115, 116, 117, 118, 119, 120gsumvalx 18539 . 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 2733 . . 3 (Base‘𝐻) = (Base‘𝐻)
123 eqid 2733 . . 3 (0g𝐻) = (0g𝐻)
124 eqid 2733 . . 3 (+g𝐻) = (+g𝐻)
125 eqid 2733 . . 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 18539 . 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 2783 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  cdif 3911  wss 3914  c0 4286  ifcif 4490  ccnv 5636  dom cdm 5637  ran crn 5638  cima 5640  ccom 5641  cio 6450  Fun wfun 6494   Fn wfn 6495  wf 6496  1-1-ontowf1o 6499  cfv 6500  (class class class)co 7361  1c1 11060  cz 12507  cuz 12771  ...cfz 13433  seqcseq 13915  chash 14239  Basecbs 17091  +gcplusg 17141  0gc0g 17329   Σg cgsu 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-seq 13916  df-0g 17331  df-gsum 17332
This theorem is referenced by:  gsumpropd2  18543
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