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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoneglmul | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.) |
Ref | Expression |
---|---|
ringnegmul.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringnegmul.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ringnegmul.3 | ⊢ 𝑋 = ran 𝐺 |
ringnegmul.4 | ⊢ 𝑁 = (inv‘𝐺) |
Ref | Expression |
---|---|
rngoneglmul | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegmul.3 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
2 | ringnegmul.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | 2 | rneqi 5802 | . . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
4 | 1, 3 | eqtri 2844 | . . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅) |
5 | ringnegmul.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | eqid 2821 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
7 | 4, 5, 6 | rngo1cl 35211 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋) |
8 | ringnegmul.4 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
9 | 2, 1, 8 | rngonegcl 35199 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
10 | 7, 9 | mpdan 685 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
11 | 2, 5, 1 | rngoass 35178 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
12 | 11 | 3exp2 1350 | . . . 4 ⊢ (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))))) |
13 | 10, 12 | mpd 15 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))))) |
14 | 13 | 3imp 1107 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
15 | 2, 5, 1, 8, 6 | rngonegmn1l 35213 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
16 | 15 | 3adant3 1128 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
17 | 16 | oveq1d 7165 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵)) |
18 | 2, 5, 1 | rngocl 35173 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
19 | 18 | 3expb 1116 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋) |
20 | 2, 5, 1, 8, 6 | rngonegmn1l 35213 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
21 | 19, 20 | syldan 593 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
22 | 21 | 3impb 1111 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
23 | 14, 17, 22 | 3eqtr4rd 2867 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ran crn 5551 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 GIdcgi 28261 invcgn 28262 RingOpscrngo 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 |
This theorem is referenced by: rngosubdir 35218 |
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