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

Theorem ringpropd 18628
Description: If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ringpropd.1 (𝜑𝐵 = (Base‘𝐾))
ringpropd.2 (𝜑𝐵 = (Base‘𝐿))
ringpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
ringpropd.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
ringpropd (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem ringpropd
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 805 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝜑)
2 simprll 819 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢𝐵)
3 simplrl 817 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐾 ∈ Grp)
4 simprlr 820 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣𝐵)
5 ringpropd.1 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = (Base‘𝐾))
65ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐵 = (Base‘𝐾))
74, 6eleqtrd 2732 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣 ∈ (Base‘𝐾))
8 simprr 811 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤𝐵)
98, 6eleqtrd 2732 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤 ∈ (Base‘𝐾))
10 eqid 2651 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
11 eqid 2651 . . . . . . . . . . . . . . . 16 (+g𝐾) = (+g𝐾)
1210, 11grpcl 17477 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
133, 7, 9, 12syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
1413, 6eleqtrrd 2733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) ∈ 𝐵)
15 ringpropd.4 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
1615oveqrspc2v 6713 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝐵 ∧ (𝑣(+g𝐾)𝑤) ∈ 𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)))
171, 2, 14, 16syl12anc 1364 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)))
18 ringpropd.3 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1918oveqrspc2v 6713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
201, 4, 8, 19syl12anc 1364 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
2120oveq2d 6706 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
2217, 21eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
23 simplrr 818 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
242, 6eleqtrd 2732 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢 ∈ (Base‘𝐾))
25 eqid 2651 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝐾) = (mulGrp‘𝐾)
2625, 10mgpbas 18541 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27 eqid 2651 . . . . . . . . . . . . . . . . 17 (.r𝐾) = (.r𝐾)
2825, 27mgpplusg 18539 . . . . . . . . . . . . . . . 16 (.r𝐾) = (+g‘(mulGrp‘𝐾))
2926, 28mndcl 17348 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3023, 24, 7, 29syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3130, 6eleqtrrd 2733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ 𝐵)
3226, 28mndcl 17348 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3323, 24, 9, 32syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3433, 6eleqtrrd 2733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ 𝐵)
3518oveqrspc2v 6713 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(.r𝐾)𝑣) ∈ 𝐵 ∧ (𝑢(.r𝐾)𝑤) ∈ 𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)))
361, 31, 34, 35syl12anc 1364 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)))
3715oveqrspc2v 6713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
381, 2, 4, 37syl12anc 1364 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
3915oveqrspc2v 6713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
401, 2, 8, 39syl12anc 1364 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
4138, 40oveq12d 6708 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)))
4236, 41eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)))
4322, 42eqeq12d 2666 . . . . . . . . . 10 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ↔ (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤))))
4410, 11grpcl 17477 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
453, 24, 7, 44syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
4645, 6eleqtrrd 2733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
4715oveqrspc2v 6713 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(+g𝐾)𝑣) ∈ 𝐵𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
481, 46, 8, 47syl12anc 1364 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
4918oveqrspc2v 6713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
501, 2, 4, 49syl12anc 1364 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
5150oveq1d 6705 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5248, 51eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5326, 28mndcl 17348 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5423, 7, 9, 53syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5554, 6eleqtrrd 2733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ 𝐵)
5618oveqrspc2v 6713 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(.r𝐾)𝑤) ∈ 𝐵 ∧ (𝑣(.r𝐾)𝑤) ∈ 𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)))
571, 34, 55, 56syl12anc 1364 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)))
5815oveqrspc2v 6713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
591, 4, 8, 58syl12anc 1364 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
6040, 59oveq12d 6708 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))
6157, 60eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))
6252, 61eqeq12d 2666 . . . . . . . . . 10 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) ↔ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))))
6343, 62anbi12d 747 . . . . . . . . 9 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
6463anassrs 681 . . . . . . . 8 ((((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ (𝑢𝐵𝑣𝐵)) ∧ 𝑤𝐵) → (((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
6564ralbidva 3014 . . . . . . 7 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ (𝑢𝐵𝑣𝐵)) → (∀𝑤𝐵 ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑤𝐵 ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
66652ralbidva 3017 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑢𝐵𝑣𝐵𝑤𝐵 ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑢𝐵𝑣𝐵𝑤𝐵 ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
675adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐾))
6867raleqdv 3174 . . . . . . . 8 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑤𝐵 ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))))
6967, 68raleqbidv 3182 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑣𝐵𝑤𝐵 ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))))
7067, 69raleqbidv 3182 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑢𝐵𝑣𝐵𝑤𝐵 ((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))))
71 ringpropd.2 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐿))
7271adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐿))
7372raleqdv 3174 . . . . . . . 8 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑤𝐵 ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
7472, 73raleqbidv 3182 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑣𝐵𝑤𝐵 ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
7572, 74raleqbidv 3182 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑢𝐵𝑣𝐵𝑤𝐵 ((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
7666, 70, 753bitr3d 298 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
7776pm5.32da 674 . . . 4 (𝜑 → (((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))) ↔ ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))))))
78 df-3an 1056 . . . 4 ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))) ↔ ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))))
79 df-3an 1056 . . . 4 ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))) ↔ ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
8077, 78, 793bitr4g 303 . . 3 (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))) ↔ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))))))
815, 71, 18grppropd 17484 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
825, 26syl6eq 2701 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐾)))
83 eqid 2651 . . . . . . 7 (mulGrp‘𝐿) = (mulGrp‘𝐿)
84 eqid 2651 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
8583, 84mgpbas 18541 . . . . . 6 (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
8671, 85syl6eq 2701 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐿)))
8728oveqi 6703 . . . . . 6 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88 eqid 2651 . . . . . . . 8 (.r𝐿) = (.r𝐿)
8983, 88mgpplusg 18539 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
9089oveqi 6703 . . . . . 6 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
9115, 87, 903eqtr3g 2708 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
9282, 86, 91mndpropd 17363 . . . 4 (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
9381, 923anbi12d 1440 . . 3 (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))) ↔ (𝐿 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))))))
9480, 93bitrd 268 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))) ↔ (𝐿 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤))))))
9510, 25, 11, 27isring 18597 . 2 (𝐾 ∈ Ring ↔ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) ∧ ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)))))
96 eqid 2651 . . 3 (+g𝐿) = (+g𝐿)
9784, 83, 96, 88isring 18597 . 2 (𝐿 ∈ Ring ↔ (𝐿 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)) = ((𝑢(.r𝐿)𝑣)(+g𝐿)(𝑢(.r𝐿)𝑤)) ∧ ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤) = ((𝑢(.r𝐿)𝑤)(+g𝐿)(𝑣(.r𝐿)𝑤)))))
9894, 95, 973bitr4g 303 1 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Mndcmnd 17341  Grpcgrp 17469  mulGrpcmgp 18535  Ringcrg 18593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-mgp 18536  df-ring 18595
This theorem is referenced by:  crngpropd  18629  ringprop  18630  opprringb  18678  drngpropd  18822  subrgpropd  18862  rhmpropd  18863  abvpropd  18890  lmodprop2d  18973  sraassa  19373  assapropd  19375  subrgpsr  19467  opsrring  19663
  Copyright terms: Public domain W3C validator