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Theorem lmodprop2d 18694
Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 18695 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Hypotheses
Ref Expression
lmodprop2d.b1 (𝜑𝐵 = (Base‘𝐾))
lmodprop2d.b2 (𝜑𝐵 = (Base‘𝐿))
lmodprop2d.f 𝐹 = (Scalar‘𝐾)
lmodprop2d.g 𝐺 = (Scalar‘𝐿)
lmodprop2d.p1 (𝜑𝑃 = (Base‘𝐹))
lmodprop2d.p2 (𝜑𝑃 = (Base‘𝐺))
lmodprop2d.1 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
lmodprop2d.2 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))
lmodprop2d.3 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(.r𝐹)𝑦) = (𝑥(.r𝐺)𝑦))
lmodprop2d.4 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
Assertion
Ref Expression
lmodprop2d (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑃,𝑦

Proof of Theorem lmodprop2d
Dummy variables 𝑟 𝑞 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodgrp 18639 . . . 4 (𝐾 ∈ LMod → 𝐾 ∈ Grp)
21a1i 11 . . 3 (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3 eqid 2609 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 eqid 2609 . . . . . 6 (+g𝐾) = (+g𝐾)
5 eqid 2609 . . . . . 6 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6 lmodprop2d.f . . . . . 6 𝐹 = (Scalar‘𝐾)
7 eqid 2609 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
8 eqid 2609 . . . . . 6 (+g𝐹) = (+g𝐹)
9 eqid 2609 . . . . . 6 (.r𝐹) = (.r𝐹)
10 eqid 2609 . . . . . 6 (1r𝐹) = (1r𝐹)
113, 4, 5, 6, 7, 8, 9, 10islmod 18636 . . . . 5 (𝐾 ∈ LMod ↔ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
1211simp2bi 1069 . . . 4 (𝐾 ∈ LMod → 𝐹 ∈ Ring)
1312a1i 11 . . 3 (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14 simplr 787 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐾 ∈ LMod)
15 simprl 789 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥𝑃)
16 lmodprop2d.p1 . . . . . . . . 9 (𝜑𝑃 = (Base‘𝐹))
1716ad2antrr 757 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑃 = (Base‘𝐹))
1815, 17eleqtrd 2689 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥 ∈ (Base‘𝐹))
19 simprr 791 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦𝐵)
20 lmodprop2d.b1 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐾))
2120ad2antrr 757 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐵 = (Base‘𝐾))
2219, 21eleqtrd 2689 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦 ∈ (Base‘𝐾))
233, 6, 5, 7lmodvscl 18649 . . . . . . 7 ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
2414, 18, 22, 23syl3anc 1317 . . . . . 6 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
2524, 21eleqtrrd 2690 . . . . 5 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
2625ralrimivva 2953 . . . 4 ((𝜑𝐾 ∈ LMod) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
2726ex 448 . . 3 (𝜑 → (𝐾 ∈ LMod → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵))
282, 13, 273jcad 1235 . 2 (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)))
29 lmodgrp 18639 . . . 4 (𝐿 ∈ LMod → 𝐿 ∈ Grp)
30 lmodprop2d.b2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
31 lmodprop2d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
3220, 30, 31grppropd 17206 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
3329, 32syl5ibr 234 . . 3 (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34 eqid 2609 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
35 eqid 2609 . . . . . 6 (+g𝐿) = (+g𝐿)
36 eqid 2609 . . . . . 6 ( ·𝑠𝐿) = ( ·𝑠𝐿)
37 lmodprop2d.g . . . . . 6 𝐺 = (Scalar‘𝐿)
38 eqid 2609 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
39 eqid 2609 . . . . . 6 (+g𝐺) = (+g𝐺)
40 eqid 2609 . . . . . 6 (.r𝐺) = (.r𝐺)
41 eqid 2609 . . . . . 6 (1r𝐺) = (1r𝐺)
4234, 35, 36, 37, 38, 39, 40, 41islmod 18636 . . . . 5 (𝐿 ∈ LMod ↔ (𝐿 ∈ Grp ∧ 𝐺 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
4342simp2bi 1069 . . . 4 (𝐿 ∈ LMod → 𝐺 ∈ Ring)
44 lmodprop2d.p2 . . . . 5 (𝜑𝑃 = (Base‘𝐺))
45 lmodprop2d.2 . . . . 5 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))
46 lmodprop2d.3 . . . . 5 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(.r𝐹)𝑦) = (𝑥(.r𝐺)𝑦))
4716, 44, 45, 46ringpropd 18351 . . . 4 (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
4843, 47syl5ibr 234 . . 3 (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49 simplr 787 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐿 ∈ LMod)
50 simprl 789 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥𝑃)
5144ad2antrr 757 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑃 = (Base‘𝐺))
5250, 51eleqtrd 2689 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥 ∈ (Base‘𝐺))
53 simprr 791 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦𝐵)
5430ad2antrr 757 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐵 = (Base‘𝐿))
5553, 54eleqtrd 2689 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦 ∈ (Base‘𝐿))
5634, 37, 36, 38lmodvscl 18649 . . . . . . 7 ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·𝑠𝐿)𝑦) ∈ (Base‘𝐿))
5749, 52, 55, 56syl3anc 1317 . . . . . 6 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐿)𝑦) ∈ (Base‘𝐿))
58 lmodprop2d.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
5958adantlr 746 . . . . . 6 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
6057, 59, 543eltr4d 2702 . . . . 5 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
6160ralrimivva 2953 . . . 4 ((𝜑𝐿 ∈ LMod) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
6261ex 448 . . 3 (𝜑 → (𝐿 ∈ LMod → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵))
6333, 48, 623jcad 1235 . 2 (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)))
6432adantr 479 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
6547adantr 479 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66 simpll 785 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝜑)
67 simprlr 798 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑟𝑃)
68 simprrr 800 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑤𝐵)
6958oveqrspc2v 6550 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑃𝑤𝐵)) → (𝑟( ·𝑠𝐾)𝑤) = (𝑟( ·𝑠𝐿)𝑤))
7066, 67, 68, 69syl12anc 1315 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑤) = (𝑟( ·𝑠𝐿)𝑤))
7170eleq1d 2671 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵))
72 simplr1 1095 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐾 ∈ Grp)
7320ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐵 = (Base‘𝐾))
7468, 73eleqtrd 2689 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑤 ∈ (Base‘𝐾))
75 simprrl 799 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑧𝐵)
7675, 73eleqtrd 2689 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑧 ∈ (Base‘𝐾))
773, 4grpcl 17199 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Grp ∧ 𝑤 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑤(+g𝐾)𝑧) ∈ (Base‘𝐾))
7872, 74, 76, 77syl3anc 1317 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) ∈ (Base‘𝐾))
7978, 73eleqtrrd 2690 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) ∈ 𝐵)
8058oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑃 ∧ (𝑤(+g𝐾)𝑧) ∈ 𝐵)) → (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐾)𝑧)))
8166, 67, 79, 80syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐾)𝑧)))
8231oveqrspc2v 6550 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐵𝑧𝐵)) → (𝑤(+g𝐾)𝑧) = (𝑤(+g𝐿)𝑧))
8366, 68, 75, 82syl12anc 1315 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) = (𝑤(+g𝐿)𝑧))
8483oveq2d 6543 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐿)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)))
8581, 84eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)))
86 simplr3 1097 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
87 ovrspc2v 6549 . . . . . . . . . . . . . . 15 (((𝑟𝑃𝑤𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)
8867, 68, 86, 87syl21anc 1316 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)
89 ovrspc2v 6549 . . . . . . . . . . . . . . 15 (((𝑟𝑃𝑧𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠𝐾)𝑧) ∈ 𝐵)
9067, 75, 86, 89syl21anc 1316 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑧) ∈ 𝐵)
9131oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)𝑧) ∈ 𝐵)) → ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑧)))
9266, 88, 90, 91syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑧)))
9358oveqrspc2v 6550 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑃𝑧𝐵)) → (𝑟( ·𝑠𝐾)𝑧) = (𝑟( ·𝑠𝐿)𝑧))
9466, 67, 75, 93syl12anc 1315 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑧) = (𝑟( ·𝑠𝐿)𝑧))
9570, 94oveq12d 6545 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)))
9692, 95eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)))
9785, 96eqeq12d 2624 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ↔ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧))))
98 simplr2 1096 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐹 ∈ Ring)
99 simprll 797 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑞𝑃)
10016ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑃 = (Base‘𝐹))
10199, 100eleqtrd 2689 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑞 ∈ (Base‘𝐹))
10267, 100eleqtrd 2689 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑟 ∈ (Base‘𝐹))
1037, 8ringacl 18347 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+g𝐹)𝑟) ∈ (Base‘𝐹))
10498, 101, 102, 103syl3anc 1317 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) ∈ (Base‘𝐹))
105104, 100eleqtrrd 2690 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) ∈ 𝑃)
10658oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑞(+g𝐹)𝑟) ∈ 𝑃𝑤𝐵)) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(+g𝐹)𝑟)( ·𝑠𝐿)𝑤))
10766, 105, 68, 106syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(+g𝐹)𝑟)( ·𝑠𝐿)𝑤))
10845oveqrspc2v 6550 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑟𝑃)) → (𝑞(+g𝐹)𝑟) = (𝑞(+g𝐺)𝑟))
109108ad2ant2r 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) = (𝑞(+g𝐺)𝑟))
110109oveq1d 6542 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤))
111107, 110eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤))
112 ovrspc2v 6549 . . . . . . . . . . . . . . 15 (((𝑞𝑃𝑤𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑞( ·𝑠𝐾)𝑤) ∈ 𝐵)
11399, 68, 86, 112syl21anc 1316 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)𝑤) ∈ 𝐵)
11431oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑞( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)) → ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤)) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑤)))
11566, 113, 88, 114syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤)) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑤)))
11658oveqrspc2v 6550 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑤𝐵)) → (𝑞( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐿)𝑤))
11766, 99, 68, 116syl12anc 1315 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐿)𝑤))
118117, 70oveq12d 6545 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞( ·𝑠𝐾)𝑤)(+g𝐿)(𝑟( ·𝑠𝐾)𝑤)) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤)))
119115, 118eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤)) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤)))
120111, 119eqeq12d 2624 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤)) ↔ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))))
12171, 97, 1203anbi123d 1390 . . . . . . . . . 10 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ↔ ((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤)))))
1227, 9ringcl 18330 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(.r𝐹)𝑟) ∈ (Base‘𝐹))
12398, 101, 102, 122syl3anc 1317 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) ∈ (Base‘𝐹))
124123, 100eleqtrrd 2690 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) ∈ 𝑃)
12558oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑞(.r𝐹)𝑟) ∈ 𝑃𝑤𝐵)) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐹)𝑟)( ·𝑠𝐿)𝑤))
12666, 124, 68, 125syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐹)𝑟)( ·𝑠𝐿)𝑤))
12746oveqrspc2v 6550 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑟𝑃)) → (𝑞(.r𝐹)𝑟) = (𝑞(.r𝐺)𝑟))
128127ad2ant2r 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) = (𝑞(.r𝐺)𝑟))
129128oveq1d 6542 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤))
130126, 129eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤))
13158oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑞𝑃 ∧ (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)) → (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐾)𝑤)))
13266, 99, 88, 131syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐾)𝑤)))
13370oveq2d 6543 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)))
134132, 133eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)))
135130, 134eqeq12d 2624 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ↔ ((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤))))
1367, 10ringidcl 18337 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Ring → (1r𝐹) ∈ (Base‘𝐹))
13798, 136syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (1r𝐹) ∈ (Base‘𝐹))
138137, 100eleqtrrd 2690 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (1r𝐹) ∈ 𝑃)
13958oveqrspc2v 6550 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((1r𝐹) ∈ 𝑃𝑤𝐵)) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐹)( ·𝑠𝐿)𝑤))
14066, 138, 68, 139syl12anc 1315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐹)( ·𝑠𝐿)𝑤))
14116, 44, 46rngidpropd 18464 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐹) = (1r𝐺))
142141ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (1r𝐹) = (1r𝐺))
143142oveq1d 6542 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐿)𝑤) = ((1r𝐺)( ·𝑠𝐿)𝑤))
144140, 143eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐺)( ·𝑠𝐿)𝑤))
145144eqeq1d 2611 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤 ↔ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))
146135, 145anbi12d 742 . . . . . . . . . 10 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤) ↔ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)))
147121, 146anbi12d 742 . . . . . . . . 9 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
148147anassrs 677 . . . . . . . 8 ((((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ (𝑞𝑃𝑟𝑃)) ∧ (𝑧𝐵𝑤𝐵)) → ((((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
1491482ralbidva 2970 . . . . . . 7 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ (𝑞𝑃𝑟𝑃)) → (∀𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
1501492ralbidva 2970 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑞𝑃𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑞𝑃𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
15116adantr 479 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
15220adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
153152eleq2d 2672 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾)))
1541533anbi1d 1394 . . . . . . . . . . 11 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ↔ ((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤)))))
155154anbi1d 736 . . . . . . . . . 10 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
156152, 155raleqbidv 3128 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
157152, 156raleqbidv 3128 . . . . . . . 8 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
158151, 157raleqbidv 3128 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
159151, 158raleqbidv 3128 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑞𝑃𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))))
16044adantr 479 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
16130adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
162161eleq2d 2672 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿)))
1631623anbi1d 1394 . . . . . . . . . . 11 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ↔ ((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤)))))
164163anbi1d 736 . . . . . . . . . 10 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
165161, 164raleqbidv 3128 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
166161, 165raleqbidv 3128 . . . . . . . 8 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)) ↔ ∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
167160, 166raleqbidv 3128 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)) ↔ ∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
168160, 167raleqbidv 3128 . . . . . 6 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑞𝑃𝑟𝑃𝑧𝐵𝑤𝐵 (((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
169150, 159, 1683bitr3d 296 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))))
17064, 65, 1693anbi123d 1390 . . . 4 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = ((𝑟( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑧)) ∧ ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞( ·𝑠𝐾)𝑤)(+g𝐾)(𝑟( ·𝑠𝐾)𝑤))) ∧ (((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) ∧ ((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤))) ↔ (𝐿 ∈ Grp ∧ 𝐺 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·𝑠𝐿)(𝑤(+g𝐿)𝑧)) = ((𝑟( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑧)) ∧ ((𝑞(+g𝐺)𝑟)( ·𝑠𝐿)𝑤) = ((𝑞( ·𝑠𝐿)𝑤)(+g𝐿)(𝑟( ·𝑠𝐿)𝑤))) ∧ (((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐿)𝑤)) ∧ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤)))))
171170, 11, 423bitr4g 301 . . 3 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
172171ex 448 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)))
17328, 63, 172pm5.21ndd 367 1 (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  .rcmulr 15715  Scalarcsca 15717   ·𝑠 cvsca 15718  Grpcgrp 17191  1rcur 18270  Ringcrg 18316  LModclmod 18632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-plusg 15727  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-mgp 18259  df-ur 18271  df-ring 18318  df-lmod 18634
This theorem is referenced by:  lmodpropd  18695  lvecprop2d  18933
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