MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumpropd2lem Structured version   Visualization version   GIF version

Theorem gsumpropd2lem 18598
Description: Lemma for gsumpropd2 18599. (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 7436 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑎(+g𝐻)𝑏))
65oveqrspc2v 7436 . . . . . . . . . 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 3328 . . . . 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 3449 . . . 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 4015 . . 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 18581 . . 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 7079 . . . . . . . . . . . . 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 3984 . . . . . . . . . . 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 14017 . . . . . . . . . 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 3177 . . . . . 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 6528 . . . 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 4123 . . . . . . . . . . . . . . 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 6060 . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) = (♯‘𝐵))
4342fveq2d 6896 . . . . . . . . . . 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 6836 . . . . . . . . . . . . . . . 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 6838 . . . . . . . . . . . . . . . . . 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 7425 . . . . . . . . . . . . . . . . . 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 6990 . . . . . . . . . . . . . . 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 7067 . . . . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . . 19 ((𝐹 “ V) ∖ (𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)})) ⊆ (𝐹 “ V)
6361, 62eqsstrdi 4037 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ (𝐹 “ V))
64 dfdm4 5896 . . . . . . . . . . . . . . . . . . 19 dom 𝐹 = ran 𝐹
65 dfrn4 6202 . . . . . . . . . . . . . . . . . . 19 ran 𝐹 = (𝐹 “ V)
6664, 65eqtri 2761 . . . . . . . . . . . . . . . . . 18 dom 𝐹 = (𝐹 “ V)
6763, 66sseqtrrdi 4034 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ dom 𝐹)
6867ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69 f1of 6834 . . . . . . . . . . . . . . . . . 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 7088 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ 𝐴)
7368, 72sseldd 3984 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓𝑎) ∈ dom 𝐹)
74 fvelrn 7079 . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . 12 ((((𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹𝑓)‘𝑎) ∈ (Base‘𝐺))
7827caovclg 7599 . . . . . . . . . . . . 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 14017 . . . . . . . . . . 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 12592 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
84 seqfn 13978 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐺), (𝐹𝑓)) Fn (ℤ‘1))
85 fndm 6653 . . . . . . . . . . . . . . . . 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 6927 . . . . . . . . . . . . . 14 (¬ (♯‘𝐵) ∈ dom seq1((+g𝐺), (𝐹𝑓)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
9088, 89syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ‘1)) → (seq1((+g𝐺), (𝐹𝑓))‘(♯‘𝐵)) = ∅)
91 seqfn 13978 . . . . . . . . . . . . . . . . 17 (1 ∈ ℤ → seq1((+g𝐻), (𝐹𝑓)) Fn (ℤ‘1))
92 fndm 6653 . . . . . . . . . . . . . . . . 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 6927 . . . . . . . . . . . . . 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 6830 . . . . . . . . 9 (𝜑 → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐵))–1-1-onto𝐴))
10541f1oeq3d 6831 . . . . . . . . 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 6528 . . . 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 4550 . . 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 4557 . 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 18595 . 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 18595 . 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 3062  wrex 3071  {crab 3433  Vcvv 3475  cdif 3946  wss 3949  c0 4323  ifcif 4529  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680  ccom 5681  cio 6494  Fun wfun 6538   Fn wfn 6539  wf 6540  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  1c1 11111  cz 12558  cuz 12822  ...cfz 13484  seqcseq 13966  chash 14290  Basecbs 17144  +gcplusg 17197  0gc0g 17385   Σg cgsu 17386
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-seq 13967  df-0g 17387  df-gsum 17388
This theorem is referenced by:  gsumpropd2  18599
  Copyright terms: Public domain W3C validator