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Theorem ringpropd 20286
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 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 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐵 = (Base‘𝐾))
74, 6eleqtrd 2842 . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤 ∈ (Base‘𝐾))
10 eqid 2736 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
11 eqid 2736 . . . . . . . . . . . . . . . 16 (+g𝐾) = (+g𝐾)
1210, 11grpcl 18960 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
2217, 21eqtrd 2776 . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢 ∈ (Base‘𝐾))
25 eqid 2736 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝐾) = (mulGrp‘𝐾)
2625, 10mgpbas 20143 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27 eqid 2736 . . . . . . . . . . . . . . . . 17 (.r𝐾) = (.r𝐾)
2825, 27mgpplusg 20142 . . . . . . . . . . . . . . . 16 (.r𝐾) = (+g‘(mulGrp‘𝐾))
2926, 28mndcl 18756 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ 𝐵)
3226, 28mndcl 18756 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ 𝐵)
3518oveqrspc2v 7459 . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 2752 . . . . . . . . . 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 18960 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
4715oveqrspc2v 7459 . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5248, 51eqtrd 2776 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5326, 28mndcl 18756 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ 𝐵)
5618oveqrspc2v 7459 . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 2752 . . . . . . . . . 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 467 . . . . . . . 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 3175 . . . . . . 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 3218 . . . . . 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 3325 . . . . . . . 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 3345 . . . . . . 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 3345 . . . . . 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 3325 . . . . . . . 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 3345 . . . . . . 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 3345 . . . . . 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 579 . . . 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 1088 . . . 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 1088 . . . 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 18970 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
825, 26eqtrdi 2792 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐾)))
83 eqid 2736 . . . . . . 7 (mulGrp‘𝐿) = (mulGrp‘𝐿)
84 eqid 2736 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
8583, 84mgpbas 20143 . . . . . 6 (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
8671, 85eqtrdi 2792 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐿)))
8728oveqi 7445 . . . . . 6 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88 eqid 2736 . . . . . . . 8 (.r𝐿) = (.r𝐿)
8983, 88mgpplusg 20142 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
9089oveqi 7445 . . . . . 6 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
9115, 87, 903eqtr3g 2799 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
9282, 86, 91mndpropd 18773 . . . 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 20235 . 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 2736 . . 3 (+g𝐿) = (+g𝐿)
9784, 83, 96, 88isring 20235 . 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  Mndcmnd 18748  Grpcgrp 18952  mulGrpcmgp 20138  Ringcrg 20231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-mgp 20139  df-ring 20233
This theorem is referenced by:  crngpropd  20287  ringprop  20288  opprringb  20349  nzrpropd  20521  subrgpropd  20609  rhmpropd  20610  drngpropd  20770  abvpropd  20837  lmodprop2d  20923  sraring  21194  sraassaOLD  21891  assapropd  21893  subrgpsr  21999  opsrring  22247
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