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Theorem lmodprop2d 19148
 Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 19149 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 19093 . . . 4 (𝐾 ∈ LMod → 𝐾 ∈ Grp)
21a1i 11 . . 3 (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3 eqid 2761 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 eqid 2761 . . . . . 6 (+g𝐾) = (+g𝐾)
5 eqid 2761 . . . . . 6 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6 lmodprop2d.f . . . . . 6 𝐹 = (Scalar‘𝐾)
7 eqid 2761 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
8 eqid 2761 . . . . . 6 (+g𝐹) = (+g𝐹)
9 eqid 2761 . . . . . 6 (.r𝐹) = (.r𝐹)
10 eqid 2761 . . . . . 6 (1r𝐹) = (1r𝐹)
113, 4, 5, 6, 7, 8, 9, 10islmod 19090 . . . . 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 1141 . . . 4 (𝐾 ∈ LMod → 𝐹 ∈ Ring)
1312a1i 11 . . 3 (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14 simplr 809 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐾 ∈ LMod)
15 simprl 811 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥𝑃)
16 lmodprop2d.p1 . . . . . . . . 9 (𝜑𝑃 = (Base‘𝐹))
1716ad2antrr 764 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑃 = (Base‘𝐹))
1815, 17eleqtrd 2842 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥 ∈ (Base‘𝐹))
19 simprr 813 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦𝐵)
20 lmodprop2d.b1 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐾))
2120ad2antrr 764 . . . . . . . 8 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐵 = (Base‘𝐾))
2219, 21eleqtrd 2842 . . . . . . 7 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦 ∈ (Base‘𝐾))
233, 6, 5, 7lmodvscl 19103 . . . . . . 7 ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
2414, 18, 22, 23syl3anc 1477 . . . . . 6 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
2524, 21eleqtrrd 2843 . . . . 5 (((𝜑𝐾 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
2625ralrimivva 3110 . . . 4 ((𝜑𝐾 ∈ LMod) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
2726ex 449 . . 3 (𝜑 → (𝐾 ∈ LMod → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵))
282, 13, 273jcad 1124 . 2 (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)))
29 lmodgrp 19093 . . . 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 17659 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
3329, 32syl5ibr 236 . . 3 (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34 eqid 2761 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
35 eqid 2761 . . . . . 6 (+g𝐿) = (+g𝐿)
36 eqid 2761 . . . . . 6 ( ·𝑠𝐿) = ( ·𝑠𝐿)
37 lmodprop2d.g . . . . . 6 𝐺 = (Scalar‘𝐿)
38 eqid 2761 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
39 eqid 2761 . . . . . 6 (+g𝐺) = (+g𝐺)
40 eqid 2761 . . . . . 6 (.r𝐺) = (.r𝐺)
41 eqid 2761 . . . . . 6 (1r𝐺) = (1r𝐺)
4234, 35, 36, 37, 38, 39, 40, 41islmod 19090 . . . . 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 1141 . . . 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 18803 . . . 4 (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
4843, 47syl5ibr 236 . . 3 (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49 simplr 809 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐿 ∈ LMod)
50 simprl 811 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥𝑃)
5144ad2antrr 764 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑃 = (Base‘𝐺))
5250, 51eleqtrd 2842 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑥 ∈ (Base‘𝐺))
53 simprr 813 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦𝐵)
5430ad2antrr 764 . . . . . . . 8 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝐵 = (Base‘𝐿))
5553, 54eleqtrd 2842 . . . . . . 7 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → 𝑦 ∈ (Base‘𝐿))
5634, 37, 36, 38lmodvscl 19103 . . . . . . 7 ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·𝑠𝐿)𝑦) ∈ (Base‘𝐿))
5749, 52, 55, 56syl3anc 1477 . . . . . 6 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐿)𝑦) ∈ (Base‘𝐿))
58 lmodprop2d.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
5958adantlr 753 . . . . . 6 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
6057, 59, 543eltr4d 2855 . . . . 5 (((𝜑𝐿 ∈ LMod) ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
6160ralrimivva 3110 . . . 4 ((𝜑𝐿 ∈ LMod) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
6261ex 449 . . 3 (𝜑 → (𝐿 ∈ LMod → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵))
6333, 48, 623jcad 1124 . 2 (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)))
6432adantr 472 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
6547adantr 472 . . . . 5 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66 simpll 807 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝜑)
67 simprlr 822 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑟𝑃)
68 simprrr 824 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑤𝐵)
6958oveqrspc2v 6838 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑃𝑤𝐵)) → (𝑟( ·𝑠𝐾)𝑤) = (𝑟( ·𝑠𝐿)𝑤))
7066, 67, 68, 69syl12anc 1475 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑤) = (𝑟( ·𝑠𝐿)𝑤))
7170eleq1d 2825 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵))
72 simplr1 1261 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐾 ∈ Grp)
7320ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐵 = (Base‘𝐾))
7468, 73eleqtrd 2842 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑤 ∈ (Base‘𝐾))
75 simprrl 823 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑧𝐵)
7675, 73eleqtrd 2842 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑧 ∈ (Base‘𝐾))
773, 4grpcl 17652 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Grp ∧ 𝑤 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑤(+g𝐾)𝑧) ∈ (Base‘𝐾))
7872, 74, 76, 77syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) ∈ (Base‘𝐾))
7978, 73eleqtrrd 2843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) ∈ 𝐵)
8058oveqrspc2v 6838 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑃 ∧ (𝑤(+g𝐾)𝑧) ∈ 𝐵)) → (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐾)𝑧)))
8166, 67, 79, 80syl12anc 1475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)(𝑤(+g𝐾)𝑧)) = (𝑟( ·𝑠𝐿)(𝑤(+g𝐾)𝑧)))
8231oveqrspc2v 6838 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐵𝑧𝐵)) → (𝑤(+g𝐾)𝑧) = (𝑤(+g𝐿)𝑧))
8366, 68, 75, 82syl12anc 1475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑤(+g𝐾)𝑧) = (𝑤(+g𝐿)𝑧))
8483oveq2d 6831 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)
87 ovrspc2v 6837 . . . . . . . . . . . . . . 15 (((𝑟𝑃𝑤𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)
8867, 68, 86, 87syl21anc 1476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)
89 ovrspc2v 6837 . . . . . . . . . . . . . . 15 (((𝑟𝑃𝑧𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠𝐾)𝑧) ∈ 𝐵)
9067, 75, 86, 89syl21anc 1476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑧) ∈ 𝐵)
9131oveqrspc2v 6838 . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . 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 6838 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑃𝑧𝐵)) → (𝑟( ·𝑠𝐾)𝑧) = (𝑟( ·𝑠𝐿)𝑧))
9466, 67, 75, 93syl12anc 1475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑟( ·𝑠𝐾)𝑧) = (𝑟( ·𝑠𝐿)𝑧))
9570, 94oveq12d 6833 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 1263 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝐹 ∈ Ring)
99 simprll 821 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑞𝑃)
10016ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑃 = (Base‘𝐹))
10199, 100eleqtrd 2842 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑞 ∈ (Base‘𝐹))
10267, 100eleqtrd 2842 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → 𝑟 ∈ (Base‘𝐹))
1037, 8ringacl 18799 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+g𝐹)𝑟) ∈ (Base‘𝐹))
10498, 101, 102, 103syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) ∈ (Base‘𝐹))
105104, 100eleqtrrd 2843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) ∈ 𝑃)
10658oveqrspc2v 6838 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑞(+g𝐹)𝑟) ∈ 𝑃𝑤𝐵)) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(+g𝐹)𝑟)( ·𝑠𝐿)𝑤))
10766, 105, 68, 106syl12anc 1475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(+g𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(+g𝐹)𝑟)( ·𝑠𝐿)𝑤))
10845oveqrspc2v 6838 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑟𝑃)) → (𝑞(+g𝐹)𝑟) = (𝑞(+g𝐺)𝑟))
109108ad2ant2r 800 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(+g𝐹)𝑟) = (𝑞(+g𝐺)𝑟))
110109oveq1d 6830 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 6837 . . . . . . . . . . . . . . 15 (((𝑞𝑃𝑤𝐵) ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝑞( ·𝑠𝐾)𝑤) ∈ 𝐵)
11399, 68, 86, 112syl21anc 1476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)𝑤) ∈ 𝐵)
11431oveqrspc2v 6838 . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . 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 6838 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑤𝐵)) → (𝑞( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐿)𝑤))
11766, 99, 68, 116syl12anc 1475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)𝑤) = (𝑞( ·𝑠𝐿)𝑤))
118117, 70oveq12d 6833 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 1548 . . . . . . . . . 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 18782 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(.r𝐹)𝑟) ∈ (Base‘𝐹))
12398, 101, 102, 122syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) ∈ (Base‘𝐹))
124123, 100eleqtrrd 2843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) ∈ 𝑃)
12558oveqrspc2v 6838 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑞(.r𝐹)𝑟) ∈ 𝑃𝑤𝐵)) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐹)𝑟)( ·𝑠𝐿)𝑤))
12666, 124, 68, 125syl12anc 1475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐹)𝑟)( ·𝑠𝐿)𝑤))
12746oveqrspc2v 6838 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑞𝑃𝑟𝑃)) → (𝑞(.r𝐹)𝑟) = (𝑞(.r𝐺)𝑟))
128127ad2ant2r 800 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞(.r𝐹)𝑟) = (𝑞(.r𝐺)𝑟))
129128oveq1d 6830 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑞(.r𝐹)𝑟)( ·𝑠𝐾)𝑤) = ((𝑞(.r𝐺)𝑟)( ·𝑠𝐿)𝑤))
13158oveqrspc2v 6838 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑞𝑃 ∧ (𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵)) → (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐾)𝑤)))
13266, 99, 88, 131syl12anc 1475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (𝑞( ·𝑠𝐾)(𝑟( ·𝑠𝐾)𝑤)) = (𝑞( ·𝑠𝐿)(𝑟( ·𝑠𝐾)𝑤)))
13370oveq2d 6831 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 18789 . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (1r𝐹) ∈ 𝑃)
13958oveqrspc2v 6838 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((1r𝐹) ∈ 𝑃𝑤𝐵)) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐹)( ·𝑠𝐿)𝑤))
14066, 138, 68, 139syl12anc 1475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐹)( ·𝑠𝐿)𝑤))
14116, 44, 46rngidpropd 18916 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐹) = (1r𝐺))
142141ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (1r𝐹) = (1r𝐺))
143142oveq1d 6830 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐿)𝑤) = ((1r𝐺)( ·𝑠𝐿)𝑤))
144140, 143eqtrd 2795 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → ((1r𝐹)( ·𝑠𝐾)𝑤) = ((1r𝐺)( ·𝑠𝐿)𝑤))
145144eqeq1d 2763 . . . . . . . . . . 11 (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞𝑃𝑟𝑃) ∧ (𝑧𝐵𝑤𝐵))) → (((1r𝐹)( ·𝑠𝐾)𝑤) = 𝑤 ↔ ((1r𝐺)( ·𝑠𝐿)𝑤) = 𝑤))
146135, 145anbi12d 749 . . . . . . . . . 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 749 . . . . . . . . 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 683 . . . . . . . 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 3127 . . . . . . 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 3127 . . . . . 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 472 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
15220adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
153152eleq2d 2826 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·𝑠𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐾)𝑤) ∈ (Base‘𝐾)))
1541533anbi1d 1552 . . . . . . . . . . 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 743 . . . . . . . . . 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 3292 . . . . . . . . 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 3292 . . . . . . . 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 3292 . . . . . . 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 3292 . . . . . 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 472 . . . . . . 7 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
16130adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
162161eleq2d 2826 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·𝑠𝐿)𝑤) ∈ 𝐵 ↔ (𝑟( ·𝑠𝐿)𝑤) ∈ (Base‘𝐿)))
1631623anbi1d 1552 . . . . . . . . . . 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 743 . . . . . . . . . 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 3292 . . . . . . . . 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 3292 . . . . . . . 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 3292 . . . . . . 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 3292 . . . . . 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 298 . . . . 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 1548 . . . 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 303 . . 3 ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
172171ex 449 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥𝑃𝑦𝐵 (𝑥( ·𝑠𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)))
17328, 63, 172pm5.21ndd 368 1 (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ‘cfv 6050  (class class class)co 6815  Basecbs 16080  +gcplusg 16164  .rcmulr 16165  Scalarcsca 16167   ·𝑠 cvsca 16168  Grpcgrp 17644  1rcur 18722  Ringcrg 18768  LModclmod 19086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-plusg 16177  df-0g 16325  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647  df-mgp 18711  df-ur 18723  df-ring 18770  df-lmod 19088 This theorem is referenced by:  lmodpropd  19149  lvecprop2d  19389
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