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Theorem ringpropd 20302
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 767 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝜑)
2 simprll 779 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢𝐵)
3 simplrl 777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝐾 ∈ Grp)
4 simprlr 780 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑣 ∈ (Base‘𝐾))
8 simprr 773 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤𝐵)
98, 6eleqtrd 2841 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑤 ∈ (Base‘𝐾))
10 eqid 2735 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
11 eqid 2735 . . . . . . . . . . . . . . . 16 (+g𝐾) = (+g𝐾)
1210, 11grpcl 18972 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
133, 7, 9, 12syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) ∈ (Base‘𝐾))
1413, 6eleqtrrd 2842 . . . . . . . . . . . . 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 7458 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝐵 ∧ (𝑣(+g𝐾)𝑤) ∈ 𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)))
171, 2, 14, 16syl12anc 837 . . . . . . . . . . . 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 7458 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
201, 4, 8, 19syl12anc 837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(+g𝐾)𝑤) = (𝑣(+g𝐿)𝑤))
2120oveq2d 7447 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐿)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
2217, 21eqtrd 2775 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)(𝑣(+g𝐾)𝑤)) = (𝑢(.r𝐿)(𝑣(+g𝐿)𝑤)))
23 simplrr 778 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
242, 6eleqtrd 2841 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → 𝑢 ∈ (Base‘𝐾))
25 eqid 2735 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝐾) = (mulGrp‘𝐾)
2625, 10mgpbas 20158 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27 eqid 2735 . . . . . . . . . . . . . . . . 17 (.r𝐾) = (.r𝐾)
2825, 27mgpplusg 20156 . . . . . . . . . . . . . . . 16 (.r𝐾) = (+g‘(mulGrp‘𝐾))
2926, 28mndcl 18768 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3023, 24, 7, 29syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ (Base‘𝐾))
3130, 6eleqtrrd 2842 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) ∈ 𝐵)
3226, 28mndcl 18768 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3323, 24, 9, 32syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ (Base‘𝐾))
3433, 6eleqtrrd 2842 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) ∈ 𝐵)
3518oveqrspc2v 7458 . . . . . . . . . . . . 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 837 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑣)(+g𝐾)(𝑢(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑣)(+g𝐿)(𝑢(.r𝐾)𝑤)))
3715oveqrspc2v 7458 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
381, 2, 4, 37syl12anc 837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑣) = (𝑢(.r𝐿)𝑣))
3915oveqrspc2v 7458 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
401, 2, 8, 39syl12anc 837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(.r𝐾)𝑤) = (𝑢(.r𝐿)𝑤))
4138, 40oveq12d 7449 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 2751 . . . . . . . . . 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 18972 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
453, 24, 7, 44syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ (Base‘𝐾))
4645, 6eleqtrrd 2842 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) ∈ 𝐵)
4715oveqrspc2v 7458 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑢(+g𝐾)𝑣) ∈ 𝐵𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
481, 46, 8, 47syl12anc 837 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤))
4918oveqrspc2v 7458 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
501, 2, 4, 49syl12anc 837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
5150oveq1d 7446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐿)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5248, 51eqtrd 2775 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(+g𝐾)𝑣)(.r𝐾)𝑤) = ((𝑢(+g𝐿)𝑣)(.r𝐿)𝑤))
5326, 28mndcl 18768 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5423, 7, 9, 53syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ (Base‘𝐾))
5554, 6eleqtrrd 2842 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) ∈ 𝐵)
5618oveqrspc2v 7458 . . . . . . . . . . . . 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 837 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → ((𝑢(.r𝐾)𝑤)(+g𝐾)(𝑣(.r𝐾)𝑤)) = ((𝑢(.r𝐾)𝑤)(+g𝐿)(𝑣(.r𝐾)𝑤)))
5815oveqrspc2v 7458 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣𝐵𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
591, 4, 8, 58syl12anc 837 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢𝐵𝑣𝐵) ∧ 𝑤𝐵)) → (𝑣(.r𝐾)𝑤) = (𝑣(.r𝐿)𝑤))
6040, 59oveq12d 7449 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 2751 . . . . . . . . . 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 3174 . . . . . . 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 3217 . . . . . 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 3324 . . . . . . . 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 3344 . . . . . . 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 3344 . . . . . 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 3324 . . . . . . . 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 3344 . . . . . . 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 3344 . . . . . 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 18982 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
825, 26eqtrdi 2791 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐾)))
83 eqid 2735 . . . . . . 7 (mulGrp‘𝐿) = (mulGrp‘𝐿)
84 eqid 2735 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
8583, 84mgpbas 20158 . . . . . 6 (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
8671, 85eqtrdi 2791 . . . . 5 (𝜑𝐵 = (Base‘(mulGrp‘𝐿)))
8728oveqi 7444 . . . . . 6 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88 eqid 2735 . . . . . . . 8 (.r𝐿) = (.r𝐿)
8983, 88mgpplusg 20156 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
9089oveqi 7444 . . . . . 6 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
9115, 87, 903eqtr3g 2798 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
9282, 86, 91mndpropd 18785 . . . 4 (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
9381, 923anbi12d 1436 . . 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 20255 . 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 2735 . . 3 (+g𝐿) = (+g𝐿)
9784, 83, 96, 88isring 20255 . 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 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Mndcmnd 18760  Grpcgrp 18964  mulGrpcmgp 20152  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-mgp 20153  df-ring 20253
This theorem is referenced by:  crngpropd  20303  ringprop  20304  opprringb  20365  nzrpropd  20537  subrgpropd  20625  rhmpropd  20626  drngpropd  20786  abvpropd  20853  lmodprop2d  20939  sraring  21211  sraassaOLD  21908  assapropd  21910  subrgpsr  22016  opsrring  22262
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