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Theorem ringpropd 20267
Description: If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, 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 765 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝜑)
2 simprll 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢𝐵)
3 simplrl 775 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐾 ∈ Grp)
4 simprlr 778 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣𝐵)
5 ringpropd.1 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = (Base‘𝐾))
65ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐵 = (Base‘𝐾))
74, 6eleqtrd 2828 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣 ∈ (Base‘𝐾))
8 simprr 771 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤𝐵)
98, 6eleqtrd 2828 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤 ∈ (Base‘𝐾))
10 eqid 2726 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
11 eqid 2726 . . . . . . . . . . . . . . . 16 (+g𝐾) = (+g𝐾)
1210, 11grpcl 18936 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
133, 7, 9, 12syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
1413, 6eleqtrrd 2829 . . . . . . . . . . . . 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 7451 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝐵 ∧ (𝑣(+g𝐾)𝑤) ∈ 𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)))
171, 2, 14, 16syl12anc 835 . . . . . . . . . . . 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 7451 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
201, 4, 8, 19syl12anc 835 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
2120oveq2d 7440 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
2217, 21eqtrd 2766 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
23 simplrr 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
242, 6eleqtrd 2828 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢 ∈ (Base‘𝐾))
25 eqid 2726 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝐾) = (mulGrp‘𝐾)
2625, 10mgpbas 20123 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27 eqid 2726 . . . . . . . . . . . . . . . . 17 (.r𝐾) = (.r𝐾)
2825, 27mgpplusg 20121 . . . . . . . . . . . . . . . 16 (.r𝐾) = (+g‘(mulGrp‘𝐾))
2926, 28mndcl 18735 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3023, 24, 7, 29syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3130, 6eleqtrrd 2829 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ 𝐵)
3226, 28mndcl 18735 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3323, 24, 9, 32syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3433, 6eleqtrrd 2829 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ 𝐵)
3518oveqrspc2v 7451 . . . . . . . . . . . . 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 835 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)))
3715oveqrspc2v 7451 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
381, 2, 4, 37syl12anc 835 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
3915oveqrspc2v 7451 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
401, 2, 8, 39syl12anc 835 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
4138, 40oveq12d 7442 . . . . . . . . . . . 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 2766 . . . . . . . . . . 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 2742 . . . . . . . . . 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 18936 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
453, 24, 7, 44syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
4645, 6eleqtrrd 2829 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
4715oveqrspc2v 7451 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(+g𝐾)𝑣) ∈ 𝐵𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
481, 46, 8, 47syl12anc 835 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
4918oveqrspc2v 7451 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
501, 2, 4, 49syl12anc 835 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
5150oveq1d 7439 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5248, 51eqtrd 2766 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5326, 28mndcl 18735 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5423, 7, 9, 53syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5554, 6eleqtrrd 2829 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ 𝐵)
5618oveqrspc2v 7451 . . . . . . . . . . . . 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 835 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)))
5815oveqrspc2v 7451 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
591, 4, 8, 58syl12anc 835 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
6040, 59oveq12d 7442 . . . . . . . . . . . 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 2766 . . . . . . . . . . 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 2742 . . . . . . . . . 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 630 . . . . . . . . 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 466 . . . . . . . 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 3166 . . . . . . 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 479 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐾))
6867raleqdv 3315 . . . . . . . 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 3330 . . . . . . 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 3330 . . . . . 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 479 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐿))
7372raleqdv 3315 . . . . . . . 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 3330 . . . . . . 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 3330 . . . . . 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 308 . . . . 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 577 . . . 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 1086 . . . 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 1086 . . . 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 313 . . 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 18946 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
825, 26eqtrdi 2782 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐾)))
83 eqid 2726 . . . . . . 7 (mulGrp‘𝐿) = (mulGrp‘𝐿)
84 eqid 2726 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
8583, 84mgpbas 20123 . . . . . 6 (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
8671, 85eqtrdi 2782 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐿)))
8728oveqi 7437 . . . . . 6 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88 eqid 2726 . . . . . . . 8 (.r𝐿) = (.r𝐿)
8983, 88mgpplusg 20121 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
9089oveqi 7437 . . . . . 6 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
9115, 87, 903eqtr3g 2789 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
9282, 86, 91mndpropd 18752 . . . 4 (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
9381, 923anbi12d 1434 . . 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 278 . 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 20220 . 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 2726 . . 3 (+g𝐿) = (+g𝐿)
9784, 83, 96, 88isring 20220 . 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 313 1 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cfv 6554  (class class class)co 7424  Basecbs 17213  +gcplusg 17266  .rcmulr 17267  Mndcmnd 18727  Grpcgrp 18928  mulGrpcmgp 20117  Ringcrg 20216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-plusg 17279  df-0g 17456  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-mgp 20118  df-ring 20218
This theorem is referenced by:  crngpropd  20268  ringprop  20269  opprringb  20330  nzrpropd  20502  subrgpropd  20592  rhmpropd  20593  drngpropd  20747  abvpropd  20814  lmodprop2d  20900  sraring  21172  sraassaOLD  21867  assapropd  21869  subrgpsr  21987  opsrring  22234
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