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Theorem issrngd 19554
 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 2824 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2824 . . 3 (1r𝑅) = (1r𝑅)
3 eqid 2824 . . . 4 (oppr𝑅) = (oppr𝑅)
43, 2oppr1 19306 . . 3 (1r𝑅) = (1r‘(oppr𝑅))
5 eqid 2824 . . 3 (.r𝑅) = (.r𝑅)
6 eqid 2824 . . 3 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
7 issrngd.r . . 3 (𝜑𝑅 ∈ Ring)
83opprring 19303 . . . 4 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
97, 8syl 17 . . 3 (𝜑 → (oppr𝑅) ∈ Ring)
10 id 22 . . . . . . . . 9 (𝑥 = (1r𝑅) → 𝑥 = (1r𝑅))
11 fveq2 6666 . . . . . . . . . 10 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘(1r𝑅)))
1211fveq2d 6670 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
1310, 12eqeq12d 2840 . . . . . . . 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 2902 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐾𝑥 ∈ (Base‘𝑅)))
18 issrngd.c . . . . . . . . . . . . . 14 (𝜑 = (*𝑟𝑅))
1918fveq1d 6668 . . . . . . . . . . . . . 14 (𝜑 → ( 𝑥) = ((*𝑟𝑅)‘𝑥))
2018, 19fveq12d 6673 . . . . . . . . . . . . 13 (𝜑 → ( ‘( 𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2120eqeq1d 2826 . . . . . . . . . . . 12 (𝜑 → (( ‘( 𝑥)) = 𝑥 ↔ ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2215, 17, 213imtr3d 294 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥))
2322imp 407 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = 𝑥)
2423eqcomd 2830 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
2524ralrimiva 3186 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
261, 2ringidcl 19240 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
277, 26syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
2813, 25, 27rspcdva 3628 . . . . . . 7 (𝜑 → (1r𝑅) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
2928oveq1d 7166 . . . . . 6 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
3011eleq1d 2901 . . . . . . . 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 2911 . . . . . . . . . 10 (𝜑 → (( 𝑥) ∈ 𝐾 ↔ ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3432, 17, 333imtr3d 294 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅)))
3534ralrimiv 3185 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
3630, 35, 27rspcdva 3628 . . . . . . 7 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) ∈ (Base‘𝑅))
37 issrngd.dt . . . . . . . . . 10 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))
38373expib 1116 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥))))
3916eleq2d 2902 . . . . . . . . . 10 (𝜑 → (𝑦𝐾𝑦 ∈ (Base‘𝑅)))
4017, 39anbi12d 630 . . . . . . . . 9 (𝜑 → ((𝑥𝐾𝑦𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
41 issrngd.t . . . . . . . . . . . 12 (𝜑· = (.r𝑅))
4241oveqd 7168 . . . . . . . . . . 11 (𝜑 → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
4318, 42fveq12d 6673 . . . . . . . . . 10 (𝜑 → ( ‘(𝑥 · 𝑦)) = ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)))
4418fveq1d 6668 . . . . . . . . . . 11 (𝜑 → ( 𝑦) = ((*𝑟𝑅)‘𝑦))
4541, 44, 19oveq123d 7172 . . . . . . . . . 10 (𝜑 → (( 𝑦) · ( 𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
4643, 45eqeq12d 2840 . . . . . . . . 9 (𝜑 → (( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)) ↔ ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4738, 40, 463imtr3d 294 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))))
4847ralrimivv 3194 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)))
49 fvoveq1 7174 . . . . . . . . 9 (𝑥 = (1r𝑅) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)))
5011oveq2d 7167 . . . . . . . . 9 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5149, 50eqeq12d 2840 . . . . . . . 8 (𝑥 = (1r𝑅) → (((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥)) ↔ ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
52 oveq2 7159 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5352fveq2d 6670 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)𝑦)) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
54 fveq2 6666 . . . . . . . . . 10 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
5554oveq1d 7166 . . . . . . . . 9 (𝑦 = ((*𝑟𝑅)‘(1r𝑅)) → (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5653, 55eqeq12d 2840 . . . . . . . 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 3637 . . . . . . 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 835 . . . . . 6 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = (((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅)))(.r𝑅)((*𝑟𝑅)‘(1r𝑅))))
5929, 58eqtr4d 2863 . . . . 5 (𝜑 → ((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅))) = ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))))
601, 5, 2ringlidm 19243 . . . . . 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 6670 . . . . 5 (𝜑 → ((*𝑟𝑅)‘((1r𝑅)(.r𝑅)((*𝑟𝑅)‘(1r𝑅)))) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
6359, 61, 623eqtr3d 2868 . . . 4 (𝜑 → ((*𝑟𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘(1r𝑅))))
64 eqid 2824 . . . . . 6 (*𝑟𝑅) = (*𝑟𝑅)
65 eqid 2824 . . . . . 6 (*rf𝑅) = (*rf𝑅)
661, 64, 65stafval 19541 . . . . 5 ((1r𝑅) ∈ (Base‘𝑅) → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6727, 66syl 17 . . . 4 (𝜑 → ((*rf𝑅)‘(1r𝑅)) = ((*𝑟𝑅)‘(1r𝑅)))
6863, 67, 283eqtr4d 2870 . . 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 19298 . . . . 5 (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)) = (((*𝑟𝑅)‘𝑦)(.r𝑅)((*𝑟𝑅)‘𝑥))
7169, 70syl6eqr 2878 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(.r‘(oppr𝑅))((*𝑟𝑅)‘𝑦)))
721, 5ringcl 19233 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
73723expb 1114 . . . . . 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 19541 . . . . 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 19541 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑥))
781, 64, 65stafval 19541 . . . . . 6 (𝑦 ∈ (Base‘𝑅) → ((*rf𝑅)‘𝑦) = ((*𝑟𝑅)‘𝑦))
7977, 78oveqan12d 7170 . . . . 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 2870 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf𝑅)‘(𝑥(.r𝑅)𝑦)) = (((*rf𝑅)‘𝑥)(.r‘(oppr𝑅))((*rf𝑅)‘𝑦)))
823, 1opprbas 19301 . . 3 (Base‘𝑅) = (Base‘(oppr𝑅))
83 eqid 2824 . . 3 (+g𝑅) = (+g𝑅)
843, 83oppradd 19302 . . 3 (+g𝑅) = (+g‘(oppr𝑅))
8534imp 407 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑥) ∈ (Base‘𝑅))
861, 64, 65staffval 19540 . . . 4 (*rf𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑥))
8785, 86fmptd 6873 . . 3 (𝜑 → (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
88 issrngd.dp . . . . . . 7 ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))
89883expib 1116 . . . . . 6 (𝜑 → ((𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦))))
90 issrngd.p . . . . . . . . 9 (𝜑+ = (+g𝑅))
9190oveqd 7168 . . . . . . . 8 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
9218, 91fveq12d 6673 . . . . . . 7 (𝜑 → ( ‘(𝑥 + 𝑦)) = ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)))
9390, 19, 44oveq123d 7172 . . . . . . 7 (𝜑 → (( 𝑥) + ( 𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦)))
9492, 93eqeq12d 2840 . . . . . 6 (𝜑 → (( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)) ↔ ((*𝑟𝑅)‘(𝑥(+g𝑅)𝑦)) = (((*𝑟𝑅)‘𝑥)(+g𝑅)((*𝑟𝑅)‘𝑦))))
9589, 40, 943imtr3d 294 . . . . 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 19250 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
98973expb 1114 . . . . . 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 19541 . . . . 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 7170 . . . . 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 2870 . . 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 19403 . 2 (𝜑 → (*rf𝑅) ∈ (𝑅 RingHom (oppr𝑅)))
1061, 64, 65staffval 19540 . . . . . . . 8 (*rf𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟𝑅)‘𝑦))
107106fmpt 6869 . . . . . . 7 (∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf𝑅):(Base‘𝑅)⟶(Base‘𝑅))
10887, 107sylibr 235 . . . . . 6 (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
109108r19.21bi 3212 . . . . 5 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((*𝑟𝑅)‘𝑦) ∈ (Base‘𝑅))
110 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
111 fveq2 6666 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘𝑦))
112111fveq2d 6670 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
113110, 112eqeq12d 2840 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
114113rspccva 3625 . . . . . . . . 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 712 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
117 fveq2 6666 . . . . . . . 8 (𝑥 = ((*𝑟𝑅)‘𝑦) → ((*𝑟𝑅)‘𝑥) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦)))
118117eqeq2d 2835 . . . . . . 7 (𝑥 = ((*𝑟𝑅)‘𝑦) → (𝑦 = ((*𝑟𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑦))))
119116, 118syl5ibrcom 248 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) → 𝑦 = ((*𝑟𝑅)‘𝑥)))
12024adantrr 713 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
121 fveq2 6666 . . . . . . . 8 (𝑦 = ((*𝑟𝑅)‘𝑥) → ((*𝑟𝑅)‘𝑦) = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥)))
122121eqeq2d 2835 . . . . . . 7 (𝑦 = ((*𝑟𝑅)‘𝑥) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟𝑅)‘((*𝑟𝑅)‘𝑥))))
123120, 122syl5ibrcom 248 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟𝑅)‘𝑥) → 𝑥 = ((*𝑟𝑅)‘𝑦)))
124119, 123impbid 213 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟𝑅)‘𝑥)))
12586, 85, 109, 124f1ocnv2d 7391 . . . 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)))
127126, 106syl6reqr 2879 . 2 (𝜑 → (*rf𝑅) = (*rf𝑅))
1283, 65issrng 19543 . 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 1081   = wceq 1530   ∈ wcel 2106  ∀wral 3142   ↦ cmpt 5142  ◡ccnv 5552  ⟶wf 6347  –1-1-onto→wf1o 6350  ‘cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  *𝑟cstv 16559  1rcur 19173  Ringcrg 19219  opprcoppr 19294   RingHom crh 19386  *rfcstf 19536  *-Ringcsr 19537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-grp 18038  df-ghm 18288  df-mgp 19162  df-ur 19174  df-ring 19221  df-oppr 19295  df-rnghom 19389  df-staf 19538  df-srng 19539 This theorem is referenced by:  idsrngd  19555  cnsrng  20495  hlhilsrnglem  38957
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