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Theorem issrngd 19144
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 2765 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2765 . . 3 (1r𝑅) = (1r𝑅)
3 eqid 2765 . . . 4 (oppr𝑅) = (oppr𝑅)
43, 2oppr1 18915 . . 3 (1r𝑅) = (1r‘(oppr𝑅))
5 eqid 2765 . . 3 (.r𝑅) = (.r𝑅)
6 eqid 2765 . . 3 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
7 issrngd.r . . 3 (𝜑𝑅 ∈ Ring)
83opprring 18912 . . . 4 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
97, 8syl 17 . . 3 (𝜑 → (oppr𝑅) ∈ Ring)
10 id 22 . . . . . . . . 9 (𝑥 = (1r𝑅) → 𝑥 = (1r𝑅))
11 fveq2 6379 . . . . . . . . . 10 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘(1r𝑅)))
1211fveq2d 6383 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
1310, 12eqeq12d 2780 . . . . . . . 8 (𝑥 = (1r𝑅) → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))))
14 issrngd.id . . . . . . . . . . . . 13 ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)
1514ex 401 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾 → ( ‘( 𝑥)) = 𝑥))
16 issrngd.k . . . . . . . . . . . . 13 (𝜑𝐾 = (Base‘𝑅))
1716eleq2d 2830 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾𝑥 ∈ (Base‘𝑅)))
18 issrngd.c . . . . . . . . . . . . . 14 (𝜑 = (*𝑟𝑅))
1918fveq1d 6381 . . . . . . . . . . . . . 14 (𝜑 → ( 𝑥) = ((*𝑟𝑅)‘𝑥))
2018, 19fveq12d 6386 . . . . . . . . . . . . 13 (𝜑 → ( ‘( 𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2120eqeq1d 2767 . . . . . . . . . . . 12 (𝜑 → (( ‘( 𝑥)) = 𝑥 ↔ ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2215, 17, 213imtr3d 284 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2322imp 395 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥)
2423eqcomd 2771 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2524ralrimiva 3113 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
261, 2ringidcl 18849 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
277, 26syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
2813, 25, 27rspcdva 3468 . . . . . . 7 (𝜑 → (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
2928oveq1d 6861 . . . . . 6 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
3011eleq1d 2829 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅) ↔ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)))
31 issrngd.cl . . . . . . . . . . 11 ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)
3231ex 401 . . . . . . . . . 10 (𝜑 → (𝑥𝐾 → ( 𝑥) ∈ 𝐾))
3319, 16eleq12d 2838 . . . . . . . . . 10 (𝜑 → (( 𝑥) ∈ 𝐾 ↔ ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3432, 17, 333imtr3d 284 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3534ralrimiv 3112 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
3630, 35, 27rspcdva 3468 . . . . . . 7 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅))
37 issrngd.dt . . . . . . . . . 10 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))
38373expib 1152 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥))))
3916eleq2d 2830 . . . . . . . . . 10 (𝜑 → (𝑦𝐾𝑦 ∈ (Base‘𝑅)))
4017, 39anbi12d 624 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
41 issrngd.t . . . . . . . . . . . 12 (𝜑· = (.r𝑅))
4241oveqd 6863 . . . . . . . . . . 11 (𝜑 → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
4318, 42fveq12d 6386 . . . . . . . . . 10 (𝜑 → ( ‘(𝑥 · 𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
4418fveq1d 6381 . . . . . . . . . . 11 (𝜑 → ( 𝑦) = ((*𝑟𝑅)‘𝑦))
4541, 44, 19oveq123d 6867 . . . . . . . . . 10 (𝜑 → (( 𝑦) · ( 𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
4643, 45eqeq12d 2780 . . . . . . . . 9 (𝜑 → (( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)) ↔ ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4738, 40, 463imtr3d 284 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4847ralrimivv 3117 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
49 fvoveq1 6869 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)))
5011oveq2d 6862 . . . . . . . . 9 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5149, 50eqeq12d 2780 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
52 oveq2 6854 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5352fveq2d 6383 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
54 fveq2 6379 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
5554oveq1d 6861 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5653, 55eqeq12d 2780 . . . . . . . 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 3476 . . . . . . 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 866 . . . . . 6 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5929, 58eqtr4d 2802 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
601, 5, 2ringlidm 18852 . . . . . 6 ((𝑅 ∈ Ring ∧ ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
617, 36, 60syl2anc 579 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘(1r𝑅)))
6261fveq2d 6383 . . . . 5 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
6359, 61, 623eqtr3d 2807 . . . 4 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
64 eqid 2765 . . . . . 6 (*𝑟𝑅) = (*𝑟𝑅)
65 eqid 2765 . . . . . 6 (*rf𝑅) = (*rf𝑅)
661, 64, 65stafval 19131 . . . . 5 ((1r𝑅) ∈ (Base‘𝑅) → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6727, 66syl 17 . . . 4 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6863, 67, 283eqtr4d 2809 . . 3 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = (1r𝑅))
6947imp 395 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
701, 5, 3, 6opprmul 18907 . . . . 5 (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))
7169, 70syl6eqr 2817 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
721, 5ringcl 18842 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
73723expb 1149 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
747, 73sylan 575 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
751, 64, 65stafval 19131 . . . . 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 19131 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑥))
781, 64, 65stafval 19131 . . . . . 6 (𝑦 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑦) = ((*𝑟𝑅)‘𝑦))
7977, 78oveqan12d 6865 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8079adantl 473 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
8171, 76, 803eqtr4d 2809 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)))
823, 1opprbas 18910 . . 3 (Base‘𝑅) = (Base‘(oppr𝑅))
83 eqid 2765 . . 3 (+g𝑅) = (+g𝑅)
843, 83oppradd 18911 . . 3 (+g𝑅) = (+g‘(oppr𝑅))
8534imp 395 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
861, 64, 65staffval 19130 . . . 4 (*rf𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑥))
8785, 86fmptd 6578 . . 3 (𝜑 → (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
88 issrngd.dp . . . . . . 7 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))
89883expib 1152 . . . . . 6 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦))))
90 issrngd.p . . . . . . . . 9 (𝜑+ = (+g𝑅))
9190oveqd 6863 . . . . . . . 8 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
9218, 91fveq12d 6386 . . . . . . 7 (𝜑 → ( ‘(𝑥 + 𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
9390, 19, 44oveq123d 6867 . . . . . . 7 (𝜑 → (( 𝑥) + ( 𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
9492, 93eqeq12d 2780 . . . . . 6 (𝜑 → (( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)) ↔ ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9589, 40, 943imtr3d 284 . . . . 5 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9695imp 395 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
971, 83ringacl 18859 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
98973expb 1149 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
997, 98sylan 575 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
1001, 64, 65stafval 19131 . . . . 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 6865 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
103102adantl 473 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf𝑅)‘𝑥)(+g𝑅)((*rf𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
10496, 101, 1033eqtr4d 2809 . . 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 19012 . 2 (𝜑 → (*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)))
1061, 64, 65staffval 19130 . . . . . . . 8 (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))
107106fmpt 6574 . . . . . . 7 (∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
10887, 107sylibr 225 . . . . . 6 (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
109108r19.21bi 3079 . . . . 5 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
110 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
111 fveq2 6379 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑦))
112111fveq2d 6383 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
113110, 112eqeq12d 2780 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
114113rspccva 3461 . . . . . . . . 9 ((∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
11525, 114sylan 575 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
116115adantrl 707 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
117 fveq2 6379 . . . . . . . 8 (𝑥 = ((*𝑟𝑅)‘𝑦) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
118117eqeq2d 2775 . . . . . . 7 (𝑥 = ((*𝑟𝑅)‘𝑦) → (𝑦 = ((*𝑟𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
119116, 118syl5ibrcom 238 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) → 𝑦 = ((*𝑟𝑅)‘𝑥)))
12024adantrr 708 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
121 fveq2 6379 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘𝑥) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
122121eqeq2d 2775 . . . . . . 7 (𝑦 = ((*𝑟𝑅)‘𝑥) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥))))
123120, 122syl5ibrcom 238 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟𝑅)‘𝑥) → 𝑥 = ((*𝑟𝑅)‘𝑦)))
124119, 123impbid 203 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟𝑅)‘𝑥)))
12586, 85, 109, 124f1ocnv2d 7088 . . . 4 (𝜑 → ((*rf𝑅):(Base‘𝑅)–1-1-onto→(Base‘𝑅) ∧ (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))))
126125simprd 489 . . 3 (𝜑(*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦)))
127126, 106syl6reqr 2818 . 2 (𝜑 → (*rf𝑅) = (*rf𝑅))
1283, 65issrng 19133 . 2 (𝑅 ∈ *-Ring ↔ ((*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)) ∧ (*rf𝑅) = (*rf𝑅)))
129105, 127, 128sylanbrc 578 1 (𝜑𝑅 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cmpt 4890  ccnv 5278  wf 6066  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6846  Basecbs 16144  +gcplusg 16228  .rcmulr 16229  *𝑟cstv 16230  1rcur 18782  Ringcrg 18828  opprcoppr 18903   RingHom crh 18995  *rfcstf 19126  *-Ringcsr 19127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-tpos 7559  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-3 11340  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-plusg 16241  df-mulr 16242  df-0g 16382  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-mhm 17615  df-grp 17706  df-ghm 17936  df-mgp 18771  df-ur 18783  df-ring 18830  df-oppr 18904  df-rnghom 18998  df-staf 19128  df-srng 19129
This theorem is referenced by:  idsrngd  19145  cnsrng  20067  hlhilsrnglem  37930
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