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Theorem ringpropd 20011
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 766 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝜑)
2 simprll 778 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢𝐵)
3 simplrl 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐾 ∈ Grp)
4 simprlr 779 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣𝐵)
5 ringpropd.1 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = (Base‘𝐾))
65ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐵 = (Base‘𝐾))
74, 6eleqtrd 2836 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣 ∈ (Base‘𝐾))
8 simprr 772 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤𝐵)
98, 6eleqtrd 2836 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤 ∈ (Base‘𝐾))
10 eqid 2733 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
11 eqid 2733 . . . . . . . . . . . . . . . 16 (+g𝐾) = (+g𝐾)
1210, 11grpcl 18761 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
133, 7, 9, 12syl3anc 1372 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
1413, 6eleqtrrd 2837 . . . . . . . . . . . . 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 7385 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝐵 ∧ (𝑣(+g𝐾)𝑤) ∈ 𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)))
171, 2, 14, 16syl12anc 836 . . . . . . . . . . . 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 7385 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
201, 4, 8, 19syl12anc 836 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
2120oveq2d 7374 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
2217, 21eqtrd 2773 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
23 simplrr 777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
242, 6eleqtrd 2836 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢 ∈ (Base‘𝐾))
25 eqid 2733 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝐾) = (mulGrp‘𝐾)
2625, 10mgpbas 19907 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27 eqid 2733 . . . . . . . . . . . . . . . . 17 (.r𝐾) = (.r𝐾)
2825, 27mgpplusg 19905 . . . . . . . . . . . . . . . 16 (.r𝐾) = (+g‘(mulGrp‘𝐾))
2926, 28mndcl 18569 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3023, 24, 7, 29syl3anc 1372 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3130, 6eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ 𝐵)
3226, 28mndcl 18569 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3323, 24, 9, 32syl3anc 1372 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3433, 6eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ 𝐵)
3518oveqrspc2v 7385 . . . . . . . . . . . . 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 836 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)))
3715oveqrspc2v 7385 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
381, 2, 4, 37syl12anc 836 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
3915oveqrspc2v 7385 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
401, 2, 8, 39syl12anc 836 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
4138, 40oveq12d 7376 . . . . . . . . . . . 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 2773 . . . . . . . . . . 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 2749 . . . . . . . . . 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 18761 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
453, 24, 7, 44syl3anc 1372 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
4645, 6eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
4715oveqrspc2v 7385 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(+g𝐾)𝑣) ∈ 𝐵𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
481, 46, 8, 47syl12anc 836 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
4918oveqrspc2v 7385 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
501, 2, 4, 49syl12anc 836 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
5150oveq1d 7373 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5248, 51eqtrd 2773 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5326, 28mndcl 18569 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5423, 7, 9, 53syl3anc 1372 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5554, 6eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ 𝐵)
5618oveqrspc2v 7385 . . . . . . . . . . . . 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 836 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)))
5815oveqrspc2v 7385 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
591, 4, 8, 58syl12anc 836 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
6040, 59oveq12d 7376 . . . . . . . . . . . 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 2773 . . . . . . . . . . 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 2749 . . . . . . . . . 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 632 . . . . . . . . 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 469 . . . . . . . 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 3169 . . . . . . 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 3207 . . . . . 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 482 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐾))
6867raleqdv 3312 . . . . . . . 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 3318 . . . . . . 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 3318 . . . . . 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 482 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐿))
7372raleqdv 3312 . . . . . . . 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 3318 . . . . . . 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 3318 . . . . . 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 309 . . . . 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 580 . . . 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 1090 . . . 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 1090 . . . 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 314 . . 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 18770 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
825, 26eqtrdi 2789 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐾)))
83 eqid 2733 . . . . . . 7 (mulGrp‘𝐿) = (mulGrp‘𝐿)
84 eqid 2733 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
8583, 84mgpbas 19907 . . . . . 6 (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
8671, 85eqtrdi 2789 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐿)))
8728oveqi 7371 . . . . . 6 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88 eqid 2733 . . . . . . . 8 (.r𝐿) = (.r𝐿)
8983, 88mgpplusg 19905 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
9089oveqi 7371 . . . . . 6 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
9115, 87, 903eqtr3g 2796 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
9282, 86, 91mndpropd 18586 . . . 4 (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
9381, 923anbi12d 1438 . . 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 279 . 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 19973 . 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 2733 . . 3 (+g𝐿) = (+g𝐿)
9784, 83, 96, 88isring 19973 . 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 314 1 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Mndcmnd 18561  Grpcgrp 18753  mulGrpcmgp 19901  Ringcrg 19969
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-mgp 19902  df-ring 19971
This theorem is referenced by:  crngpropd  20012  ringprop  20013  opprringb  20066  drngpropd  20230  subrgpropd  20271  rhmpropd  20272  abvpropd  20315  lmodprop2d  20399  sraassa  21289  assapropd  21291  subrgpsr  21404  opsrring  21632  sraring  32341
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