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Theorem issrngd 18909
Description: Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
issrngd.k (𝜑𝐾 = (Base‘𝑅))
issrngd.p (𝜑+ = (+g𝑅))
issrngd.t (𝜑· = (.r𝑅))
issrngd.c (𝜑 = (*𝑟𝑅))
issrngd.r (𝜑𝑅 ∈ Ring)
issrngd.cl ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)
issrngd.dp ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))
issrngd.dt ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))
issrngd.id ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)
Assertion
Ref Expression
issrngd (𝜑𝑅 ∈ *-Ring)
Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   + (𝑥,𝑦)   · (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem issrngd
StepHypRef Expression
1 eqid 2651 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2651 . . 3 (1r𝑅) = (1r𝑅)
3 eqid 2651 . . . 4 (oppr𝑅) = (oppr𝑅)
43, 2oppr1 18680 . . 3 (1r𝑅) = (1r‘(oppr𝑅))
5 eqid 2651 . . 3 (.r𝑅) = (.r𝑅)
6 eqid 2651 . . 3 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
7 issrngd.r . . 3 (𝜑𝑅 ∈ Ring)
83opprring 18677 . . . 4 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
97, 8syl 17 . . 3 (𝜑 → (oppr𝑅) ∈ Ring)
101, 2ringidcl 18614 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
117, 10syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
12 issrngd.id . . . . . . . . . . . . 13 ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)
1312ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾 → ( ‘( 𝑥)) = 𝑥))
14 issrngd.k . . . . . . . . . . . . 13 (𝜑𝐾 = (Base‘𝑅))
1514eleq2d 2716 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾𝑥 ∈ (Base‘𝑅)))
16 issrngd.c . . . . . . . . . . . . . 14 (𝜑 = (*𝑟𝑅))
1716fveq1d 6231 . . . . . . . . . . . . . 14 (𝜑 → ( 𝑥) = ((*𝑟𝑅)‘𝑥))
1816, 17fveq12d 6235 . . . . . . . . . . . . 13 (𝜑 → ( ‘( 𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
1918eqeq1d 2653 . . . . . . . . . . . 12 (𝜑 → (( ‘( 𝑥)) = 𝑥 ↔ ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2013, 15, 193imtr3d 282 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2120imp 444 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥)
2221eqcomd 2657 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2322ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
24 id 22 . . . . . . . . . 10 (𝑥 = (1r𝑅) → 𝑥 = (1r𝑅))
25 fveq2 6229 . . . . . . . . . . 11 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘(1r𝑅)))
2625fveq2d 6233 . . . . . . . . . 10 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
2724, 26eqeq12d 2666 . . . . . . . . 9 (𝑥 = (1r𝑅) → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))))
2827rspcv 3336 . . . . . . . 8 ((1r𝑅) ∈ (Base‘𝑅) → (∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) → (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))))
2911, 23, 28sylc 65 . . . . . . 7 (𝜑 → (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
3029oveq1d 6705 . . . . . 6 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
31 issrngd.cl . . . . . . . . . . 11 ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)
3231ex 449 . . . . . . . . . 10 (𝜑 → (𝑥𝐾 → ( 𝑥) ∈ 𝐾))
3317, 14eleq12d 2724 . . . . . . . . . 10 (𝜑 → (( 𝑥) ∈ 𝐾 ↔ ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3432, 15, 333imtr3d 282 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3534ralrimiv 2994 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
3625eleq1d 2715 . . . . . . . . 9 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅) ↔ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)))
3736rspcv 3336 . . . . . . . 8 ((1r𝑅) ∈ (Base‘𝑅) → (∀𝑥 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅) → ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)))
3811, 35, 37sylc 65 . . . . . . 7 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅))
39 issrngd.dt . . . . . . . . . 10 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))
40393expib 1287 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥))))
4114eleq2d 2716 . . . . . . . . . 10 (𝜑 → (𝑦𝐾𝑦 ∈ (Base‘𝑅)))
4215, 41anbi12d 747 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
43 issrngd.t . . . . . . . . . . . 12 (𝜑· = (.r𝑅))
4443oveqd 6707 . . . . . . . . . . 11 (𝜑 → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
4516, 44fveq12d 6235 . . . . . . . . . 10 (𝜑 → ( ‘(𝑥 · 𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
4616fveq1d 6231 . . . . . . . . . . 11 (𝜑 → ( 𝑦) = ((*𝑟𝑅)‘𝑦))
4743, 46, 17oveq123d 6711 . . . . . . . . . 10 (𝜑 → (( 𝑦) · ( 𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
4845, 47eqeq12d 2666 . . . . . . . . 9 (𝜑 → (( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)) ↔ ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4940, 42, 483imtr3d 282 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
5049ralrimivv 2999 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
51 oveq1 6697 . . . . . . . . . 10 (𝑥 = (1r𝑅) → (𝑥(.r𝑅)𝑦) = ((1r𝑅)(.r𝑅)𝑦))
5251fveq2d 6233 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)))
5325oveq2d 6706 . . . . . . . . 9 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5452, 53eqeq12d 2666 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
55 oveq2 6698 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5655fveq2d 6233 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
57 fveq2 6229 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
5857oveq1d 6705 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5956, 58eqeq12d 2666 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
6054, 59rspc2va 3354 . . . . . . 7 ((((1r𝑅) ∈ (Base‘𝑅) ∧ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))) → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
6111, 38, 50, 60syl21anc 1365 . . . . . 6 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
6230, 61eqtr4d 2688 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
631, 5, 2ringlidm 18617 . . . . . 6 ((𝑅 ∈ Ring ∧ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
647, 38, 63syl2anc 694 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
6564fveq2d 6233 . . . . 5 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
6662, 64, 653eqtr3d 2693 . . . 4 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
67 eqid 2651 . . . . . 6 (*𝑟𝑅) = (*𝑟𝑅)
68 eqid 2651 . . . . . 6 (*rf𝑅) = (*rf𝑅)
691, 67, 68stafval 18896 . . . . 5 ((1r𝑅) ∈ (Base‘𝑅) → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
7011, 69syl 17 . . . 4 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
7166, 70, 293eqtr4d 2695 . . 3 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = (1r𝑅))
7249imp 444 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
731, 5, 3, 6opprmul 18672 . . . . 5 (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))
7472, 73syl6eqr 2703 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
751, 5ringcl 18607 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
76753expb 1285 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
777, 76sylan 487 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
781, 67, 68stafval 18896 . . . . 5 ((𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
7977, 78syl 17 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
801, 67, 68stafval 18896 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑥))
811, 67, 68stafval 18896 . . . . . 6 (𝑦 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑦) = ((*𝑟𝑅)‘𝑦))
8280, 81oveqan12d 6709 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8382adantl 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8474, 79, 833eqtr4d 2695 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)))
853, 1opprbas 18675 . . 3 (Base‘𝑅) = (Base‘(oppr𝑅))
86 eqid 2651 . . 3 (+g𝑅) = (+g𝑅)
873, 86oppradd 18676 . . 3 (+g𝑅) = (+g‘(oppr𝑅))
8834imp 444 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
891, 67, 68staffval 18895 . . . 4 (*rf𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑥))
9088, 89fmptd 6425 . . 3 (𝜑 → (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
91 issrngd.dp . . . . . . 7 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))
92913expib 1287 . . . . . 6 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦))))
93 issrngd.p . . . . . . . . 9 (𝜑+ = (+g𝑅))
9493oveqd 6707 . . . . . . . 8 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
9516, 94fveq12d 6235 . . . . . . 7 (𝜑 → ( ‘(𝑥 + 𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
9693, 17, 46oveq123d 6711 . . . . . . 7 (𝜑 → (( 𝑥) + ( 𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
9795, 96eqeq12d 2666 . . . . . 6 (𝜑 → (( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)) ↔ ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9892, 42, 973imtr3d 282 . . . . 5 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9998imp 444 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
1001, 86ringacl 18624 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
1011003expb 1285 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
1027, 101sylan 487 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
1031, 67, 68stafval 18896 . . . . 5 ((𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
104102, 103syl 17 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
10580, 81oveqan12d 6709 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
106105adantl 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
10799, 104, 1063eqtr4d 2695 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)))
1081, 2, 4, 5, 6, 7, 9, 71, 84, 85, 86, 87, 90, 107isrhmd 18777 . 2 (𝜑 → (*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)))
1091, 67, 68staffval 18895 . . . . . . . 8 (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))
110109fmpt 6421 . . . . . . 7 (∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
11190, 110sylibr 224 . . . . . 6 (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
112111r19.21bi 2961 . . . . 5 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
113 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
114 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑦))
115114fveq2d 6233 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
116113, 115eqeq12d 2666 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
117116rspccva 3339 . . . . . . . . 9 ((∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
11823, 117sylan 487 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
119118adantrl 752 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
120 fveq2 6229 . . . . . . . 8 (𝑥 = ((*𝑟𝑅)‘𝑦) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
121120eqeq2d 2661 . . . . . . 7 (𝑥 = ((*𝑟𝑅)‘𝑦) → (𝑦 = ((*𝑟𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
122119, 121syl5ibrcom 237 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) → 𝑦 = ((*𝑟𝑅)‘𝑥)))
12322adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
124 fveq2 6229 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘𝑥) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
125124eqeq2d 2661 . . . . . . 7 (𝑦 = ((*𝑟𝑅)‘𝑥) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥))))
126123, 125syl5ibrcom 237 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟𝑅)‘𝑥) → 𝑥 = ((*𝑟𝑅)‘𝑦)))
127122, 126impbid 202 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟𝑅)‘𝑥)))
12889, 88, 112, 127f1ocnv2d 6928 . . . 4 (𝜑 → ((*rf𝑅):(Base‘𝑅)–1-1-onto→(Base‘𝑅) ∧ (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))))
129128simprd 478 . . 3 (𝜑(*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦)))
130129, 109syl6reqr 2704 . 2 (𝜑 → (*rf𝑅) = (*rf𝑅))
1313, 68issrng 18898 . 2 (𝑅 ∈ *-Ring ↔ ((*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)) ∧ (*rf𝑅) = (*rf𝑅)))
132108, 130, 131sylanbrc 699 1 (𝜑𝑅 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cmpt 4762  ccnv 5142  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  *𝑟cstv 15990  1rcur 18547  Ringcrg 18593  opprcoppr 18668   RingHom crh 18760  *rfcstf 18891  *-Ringcsr 18892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-mulr 16002  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-ghm 17705  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-rnghom 18763  df-staf 18893  df-srng 18894
This theorem is referenced by:  idsrngd  18910  cnsrng  19828  hlhilsrnglem  37562
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