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

Theorem mndpropd 17584
Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
mndpropd.1 (𝜑𝐵 = (Base‘𝐾))
mndpropd.2 (𝜑𝐵 = (Base‘𝐿))
mndpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
mndpropd (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem mndpropd
Dummy variables 𝑢 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 785 . . . . . 6 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝐾 ∈ Mnd)
2 simprl 787 . . . . . . 7 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
3 mndpropd.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐾))
43ad2antrr 717 . . . . . . 7 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝐵 = (Base‘𝐾))
52, 4eleqtrd 2846 . . . . . 6 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ (Base‘𝐾))
6 simprr 789 . . . . . . 7 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
76, 4eleqtrd 2846 . . . . . 6 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ (Base‘𝐾))
8 eqid 2765 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
9 eqid 2765 . . . . . . 7 (+g𝐾) = (+g𝐾)
108, 9mndcl 17569 . . . . . 6 ((𝐾 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))
111, 5, 7, 10syl3anc 1490 . . . . 5 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))
1211, 4eleqtrrd 2847 . . . 4 (((𝜑𝐾 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) ∈ 𝐵)
1312ralrimivva 3118 . . 3 ((𝜑𝐾 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵)
1413ex 401 . 2 (𝜑 → (𝐾 ∈ Mnd → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵))
15 simplr 785 . . . . . 6 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝐿 ∈ Mnd)
16 simprl 787 . . . . . . 7 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
17 mndpropd.2 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐿))
1817ad2antrr 717 . . . . . . 7 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝐵 = (Base‘𝐿))
1916, 18eleqtrd 2846 . . . . . 6 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ (Base‘𝐿))
20 simprr 789 . . . . . . 7 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
2120, 18eleqtrd 2846 . . . . . 6 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ (Base‘𝐿))
22 eqid 2765 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
23 eqid 2765 . . . . . . 7 (+g𝐿) = (+g𝐿)
2422, 23mndcl 17569 . . . . . 6 ((𝐿 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))
2515, 19, 21, 24syl3anc 1490 . . . . 5 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))
26 mndpropd.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
2726adantlr 706 . . . . 5 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
2825, 27, 183eltr4d 2859 . . . 4 (((𝜑𝐿 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) ∈ 𝐵)
2928ralrimivva 3118 . . 3 ((𝜑𝐿 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵)
3029ex 401 . 2 (𝜑 → (𝐿 ∈ Mnd → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵))
3126oveqrspc2v 6869 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
3231adantlr 706 . . . . . . . . 9 (((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
3332eleq1d 2829 . . . . . . . 8 (((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) → ((𝑢(+g𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g𝐿)𝑣) ∈ 𝐵))
34 simplll 791 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → 𝜑)
35 simplrl 795 . . . . . . . . . . . . 13 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → 𝑢𝐵)
36 simplrr 796 . . . . . . . . . . . . 13 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → 𝑣𝐵)
37 simpllr 793 . . . . . . . . . . . . 13 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵)
38 ovrspc2v 6868 . . . . . . . . . . . . 13 (((𝑢𝐵𝑣𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
3935, 36, 37, 38syl21anc 866 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
40 simpr 477 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → 𝑤𝐵)
4126oveqrspc2v 6869 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑢(+g𝐾)𝑣) ∈ 𝐵𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(+g𝐿)𝑤))
4234, 39, 40, 41syl12anc 865 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(+g𝐿)𝑤))
4334, 35, 36, 31syl12anc 865 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
4443oveq1d 6857 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → ((𝑢(+g𝐾)𝑣)(+g𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤))
4542, 44eqtrd 2799 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤))
46 ovrspc2v 6868 . . . . . . . . . . . . 13 (((𝑣𝐵𝑤𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (𝑣(+g𝐾)𝑤) ∈ 𝐵)
4736, 40, 37, 46syl21anc 866 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑣(+g𝐾)𝑤) ∈ 𝐵)
4826oveqrspc2v 6869 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝐵 ∧ (𝑣(+g𝐾)𝑤) ∈ 𝐵)) → (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(+g𝐿)(𝑣(+g𝐾)𝑤)))
4934, 35, 47, 48syl12anc 865 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(+g𝐿)(𝑣(+g𝐾)𝑤)))
5026oveqrspc2v 6869 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
5134, 36, 40, 50syl12anc 865 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
5251oveq2d 6858 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑢(+g𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))
5349, 52eqtrd 2799 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))
5445, 53eqeq12d 2780 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) ↔ ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))))
5554ralbidva 3132 . . . . . . . 8 (((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) → (∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) ↔ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))))
5633, 55anbi12d 624 . . . . . . 7 (((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ (𝑢𝐵𝑣𝐵)) → (((𝑢(+g𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ((𝑢(+g𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
57562ralbidva 3135 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢𝐵𝑣𝐵 ((𝑢(+g𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ∀𝑢𝐵𝑣𝐵 ((𝑢(+g𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
583adantr 472 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐾))
5958eleq2d 2830 . . . . . . . . 9 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾)))
6058raleqdv 3292 . . . . . . . . 9 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))))
6159, 60anbi12d 624 . . . . . . . 8 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)))))
6258, 61raleqbidv 3300 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑣𝐵 ((𝑢(+g𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)))))
6358, 62raleqbidv 3300 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢𝐵𝑣𝐵 ((𝑢(+g𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤)))))
6417adantr 472 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐿))
6564eleq2d 2830 . . . . . . . . 9 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g𝐿)𝑣) ∈ 𝐵 ↔ (𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿)))
6664raleqdv 3292 . . . . . . . . 9 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))))
6765, 66anbi12d 624 . . . . . . . 8 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))) ↔ ((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
6864, 67raleqbidv 3300 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑣𝐵 ((𝑢(+g𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
6964, 68raleqbidv 3300 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢𝐵𝑣𝐵 ((𝑢(+g𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤𝐵 ((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
7057, 63, 693bitr3d 300 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤)))))
71 simplll 791 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → 𝜑)
72 simplr 785 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → 𝑠𝐵)
73 simpr 477 . . . . . . . . . . 11 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → 𝑢𝐵)
7426oveqrspc2v 6869 . . . . . . . . . . 11 ((𝜑 ∧ (𝑠𝐵𝑢𝐵)) → (𝑠(+g𝐾)𝑢) = (𝑠(+g𝐿)𝑢))
7571, 72, 73, 74syl12anc 865 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → (𝑠(+g𝐾)𝑢) = (𝑠(+g𝐿)𝑢))
7675eqeq1d 2767 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → ((𝑠(+g𝐾)𝑢) = 𝑢 ↔ (𝑠(+g𝐿)𝑢) = 𝑢))
7726oveqrspc2v 6869 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝐵𝑠𝐵)) → (𝑢(+g𝐾)𝑠) = (𝑢(+g𝐿)𝑠))
7871, 73, 72, 77syl12anc 865 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → (𝑢(+g𝐾)𝑠) = (𝑢(+g𝐿)𝑠))
7978eqeq1d 2767 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → ((𝑢(+g𝐾)𝑠) = 𝑢 ↔ (𝑢(+g𝐿)𝑠) = 𝑢))
8076, 79anbi12d 624 . . . . . . . 8 ((((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) ∧ 𝑢𝐵) → (((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8180ralbidva 3132 . . . . . . 7 (((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) ∧ 𝑠𝐵) → (∀𝑢𝐵 ((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ∀𝑢𝐵 ((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8281rexbidva 3196 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∃𝑠𝐵𝑢𝐵 ((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ∃𝑠𝐵𝑢𝐵 ((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8358raleqdv 3292 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢𝐵 ((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢)))
8458, 83rexeqbidv 3301 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∃𝑠𝐵𝑢𝐵 ((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢)))
8564raleqdv 3292 . . . . . . 7 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∀𝑢𝐵 ((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8664, 85rexeqbidv 3301 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∃𝑠𝐵𝑢𝐵 ((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8782, 84, 863bitr3d 300 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
8870, 87anbi12d 624 . . . 4 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → ((∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢)) ↔ (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢))))
898, 9ismnd 17565 . . . 4 (𝐾 ∈ Mnd ↔ (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g𝐾)𝑣)(+g𝐾)𝑤) = (𝑢(+g𝐾)(𝑣(+g𝐾)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g𝐾)𝑢) = 𝑢 ∧ (𝑢(+g𝐾)𝑠) = 𝑢)))
9022, 23ismnd 17565 . . . 4 (𝐿 ∈ Mnd ↔ (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g𝐿)𝑣)(+g𝐿)𝑤) = (𝑢(+g𝐿)(𝑣(+g𝐿)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g𝐿)𝑢) = 𝑢 ∧ (𝑢(+g𝐿)𝑠) = 𝑢)))
9188, 89, 903bitr4g 305 . . 3 ((𝜑 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
9291ex 401 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐾)𝑦) ∈ 𝐵 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)))
9314, 30, 92pm5.21ndd 370 1 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mndcmnd 17562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4949  ax-pow 5001
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-ov 6845  df-mgm 17510  df-sgrp 17552  df-mnd 17563
This theorem is referenced by:  mndprop  17585  mhmpropd  17609  grppropd  17706  oppgmndb  18050  cmnpropd  18468  ringpropd  18849  prdsringd  18879
  Copyright terms: Public domain W3C validator