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Theorem issrngd 20320
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 2736 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2736 . . 3 (1r𝑅) = (1r𝑅)
3 eqid 2736 . . . 4 (oppr𝑅) = (oppr𝑅)
43, 2oppr1 20063 . . 3 (1r𝑅) = (1r‘(oppr𝑅))
5 eqid 2736 . . 3 (.r𝑅) = (.r𝑅)
6 eqid 2736 . . 3 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
7 issrngd.r . . 3 (𝜑𝑅 ∈ Ring)
83opprring 20060 . . . 4 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
97, 8syl 17 . . 3 (𝜑 → (oppr𝑅) ∈ Ring)
10 id 22 . . . . . . . . 9 (𝑥 = (1r𝑅) → 𝑥 = (1r𝑅))
11 fveq2 6842 . . . . . . . . . 10 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘(1r𝑅)))
1211fveq2d 6846 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
1310, 12eqeq12d 2752 . . . . . . . 8 (𝑥 = (1r𝑅) → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))))
14 issrngd.id . . . . . . . . . . . . 13 ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)
1514ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾 → ( ‘( 𝑥)) = 𝑥))
16 issrngd.k . . . . . . . . . . . . 13 (𝜑𝐾 = (Base‘𝑅))
1716eleq2d 2823 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾𝑥 ∈ (Base‘𝑅)))
18 issrngd.c . . . . . . . . . . . . . 14 (𝜑 = (*𝑟𝑅))
1918fveq1d 6844 . . . . . . . . . . . . . 14 (𝜑 → ( 𝑥) = ((*𝑟𝑅)‘𝑥))
2018, 19fveq12d 6849 . . . . . . . . . . . . 13 (𝜑 → ( ‘( 𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2120eqeq1d 2738 . . . . . . . . . . . 12 (𝜑 → (( ‘( 𝑥)) = 𝑥 ↔ ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2215, 17, 213imtr3d 292 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2322imp 407 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥)
2423eqcomd 2742 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2524ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
261, 2ringidcl 19989 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
277, 26syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
2813, 25, 27rspcdva 3582 . . . . . . 7 (𝜑 → (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
2928oveq1d 7372 . . . . . 6 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
3011eleq1d 2822 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅) ↔ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)))
31 issrngd.cl . . . . . . . . . . 11 ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)
3231ex 413 . . . . . . . . . 10 (𝜑 → (𝑥𝐾 → ( 𝑥) ∈ 𝐾))
3319, 16eleq12d 2832 . . . . . . . . . 10 (𝜑 → (( 𝑥) ∈ 𝐾 ↔ ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3432, 17, 333imtr3d 292 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3534ralrimiv 3142 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
3630, 35, 27rspcdva 3582 . . . . . . 7 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅))
37 issrngd.dt . . . . . . . . . 10 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))
38373expib 1122 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥))))
3916eleq2d 2823 . . . . . . . . . 10 (𝜑 → (𝑦𝐾𝑦 ∈ (Base‘𝑅)))
4017, 39anbi12d 631 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
41 issrngd.t . . . . . . . . . . . 12 (𝜑· = (.r𝑅))
4241oveqd 7374 . . . . . . . . . . 11 (𝜑 → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
4318, 42fveq12d 6849 . . . . . . . . . 10 (𝜑 → ( ‘(𝑥 · 𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
4418fveq1d 6844 . . . . . . . . . . 11 (𝜑 → ( 𝑦) = ((*𝑟𝑅)‘𝑦))
4541, 44, 19oveq123d 7378 . . . . . . . . . 10 (𝜑 → (( 𝑦) · ( 𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
4643, 45eqeq12d 2752 . . . . . . . . 9 (𝜑 → (( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)) ↔ ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4738, 40, 463imtr3d 292 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4847ralrimivv 3195 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
49 fvoveq1 7380 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)))
5011oveq2d 7373 . . . . . . . . 9 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5149, 50eqeq12d 2752 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
52 oveq2 7365 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5352fveq2d 6846 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
54 fveq2 6842 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
5554oveq1d 7372 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5653, 55eqeq12d 2752 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
5751, 56rspc2va 3591 . . . . . . 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))))
5827, 36, 48, 57syl21anc 836 . . . . . 6 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5929, 58eqtr4d 2779 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
601, 5, 2ringlidm 19992 . . . . . 6 ((𝑅 ∈ Ring ∧ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
617, 36, 60syl2anc 584 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
6261fveq2d 6846 . . . . 5 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
6359, 61, 623eqtr3d 2784 . . . 4 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
64 eqid 2736 . . . . . 6 (*𝑟𝑅) = (*𝑟𝑅)
65 eqid 2736 . . . . . 6 (*rf𝑅) = (*rf𝑅)
661, 64, 65stafval 20307 . . . . 5 ((1r𝑅) ∈ (Base‘𝑅) → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6727, 66syl 17 . . . 4 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6863, 67, 283eqtr4d 2786 . . 3 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = (1r𝑅))
6947imp 407 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
701, 5, 3, 6opprmul 20052 . . . . 5 (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))
7169, 70eqtr4di 2794 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
721, 5ringcl 19981 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
73723expb 1120 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
747, 73sylan 580 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
751, 64, 65stafval 20307 . . . . 5 ((𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
7674, 75syl 17 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
771, 64, 65stafval 20307 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑥))
781, 64, 65stafval 20307 . . . . . 6 (𝑦 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑦) = ((*𝑟𝑅)‘𝑦))
7977, 78oveqan12d 7376 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8079adantl 482 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8171, 76, 803eqtr4d 2786 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)))
823, 1opprbas 20056 . . 3 (Base‘𝑅) = (Base‘(oppr𝑅))
83 eqid 2736 . . 3 (+g𝑅) = (+g𝑅)
843, 83oppradd 20058 . . 3 (+g𝑅) = (+g‘(oppr𝑅))
8534imp 407 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
861, 64, 65staffval 20306 . . . 4 (*rf𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑥))
8785, 86fmptd 7062 . . 3 (𝜑 → (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
88 issrngd.dp . . . . . . 7 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))
89883expib 1122 . . . . . 6 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦))))
90 issrngd.p . . . . . . . . 9 (𝜑+ = (+g𝑅))
9190oveqd 7374 . . . . . . . 8 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
9218, 91fveq12d 6849 . . . . . . 7 (𝜑 → ( ‘(𝑥 + 𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
9390, 19, 44oveq123d 7378 . . . . . . 7 (𝜑 → (( 𝑥) + ( 𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
9492, 93eqeq12d 2752 . . . . . 6 (𝜑 → (( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)) ↔ ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9589, 40, 943imtr3d 292 . . . . 5 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9695imp 407 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
971, 83ringacl 19999 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
98973expb 1120 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
997, 98sylan 580 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
1001, 64, 65stafval 20307 . . . . 5 ((𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
10199, 100syl 17 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
10277, 78oveqan12d 7376 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
103102adantl 482 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
10496, 101, 1033eqtr4d 2786 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)))
1051, 2, 4, 5, 6, 7, 9, 68, 81, 82, 83, 84, 87, 104isrhmd 20161 . 2 (𝜑 → (*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)))
1061, 64, 65staffval 20306 . . 3 (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))
107106fmpt 7058 . . . . . . 7 (∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
10887, 107sylibr 233 . . . . . 6 (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
109108r19.21bi 3234 . . . . 5 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
110 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
111 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑦))
112111fveq2d 6846 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
113110, 112eqeq12d 2752 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
114113rspccva 3580 . . . . . . . . 9 ((∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
11525, 114sylan 580 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
116115adantrl 714 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
117 fveq2 6842 . . . . . . . 8 (𝑥 = ((*𝑟𝑅)‘𝑦) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
118117eqeq2d 2747 . . . . . . 7 (𝑥 = ((*𝑟𝑅)‘𝑦) → (𝑦 = ((*𝑟𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
119116, 118syl5ibrcom 246 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) → 𝑦 = ((*𝑟𝑅)‘𝑥)))
12024adantrr 715 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
121 fveq2 6842 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘𝑥) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
122121eqeq2d 2747 . . . . . . 7 (𝑦 = ((*𝑟𝑅)‘𝑥) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥))))
123120, 122syl5ibrcom 246 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟𝑅)‘𝑥) → 𝑥 = ((*𝑟𝑅)‘𝑦)))
124119, 123impbid 211 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟𝑅)‘𝑥)))
12586, 85, 109, 124f1ocnv2d 7606 . . . 4 (𝜑 → ((*rf𝑅):(Base‘𝑅)–1-1-onto→(Base‘𝑅) ∧ (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))))
126125simprd 496 . . 3 (𝜑(*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦)))
127106, 126eqtr4id 2795 . 2 (𝜑 → (*rf𝑅) = (*rf𝑅))
1283, 65issrng 20309 . 2 (𝑅 ∈ *-Ring ↔ ((*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)) ∧ (*rf𝑅) = (*rf𝑅)))
129105, 127, 128sylanbrc 583 1 (𝜑𝑅 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cmpt 5188  ccnv 5632  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  *𝑟cstv 17135  1rcur 19913  Ringcrg 19964  opprcoppr 20048   RingHom crh 20143  *rfcstf 20302  *-Ringcsr 20303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-rnghom 20146  df-staf 20304  df-srng 20305
This theorem is referenced by:  idsrngd  20321  cnsrng  20831  hlhilsrnglem  40420
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