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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 5800 | . . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
4 | 1, 3 | eqtri 2841 | . . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅) |
5 | ringnegmul.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | eqid 2818 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
7 | 4, 5, 6 | rngo1cl 35098 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋) |
8 | ringnegmul.4 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
9 | 2, 1, 8 | rngonegcl 35086 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
10 | 7, 9 | mpdan 683 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
11 | 2, 5, 1 | rngoass 35065 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
12 | 11 | 3exp2 1346 | . . . 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 1103 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
15 | 2, 5, 1, 8, 6 | rngonegmn1l 35100 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
16 | 15 | 3adant3 1124 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
17 | 16 | oveq1d 7160 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵)) |
18 | 2, 5, 1 | rngocl 35060 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
19 | 18 | 3expb 1112 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋) |
20 | 2, 5, 1, 8, 6 | rngonegmn1l 35100 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
21 | 19, 20 | syldan 591 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
22 | 21 | 3impb 1107 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
23 | 14, 17, 22 | 3eqtr4rd 2864 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ran crn 5549 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 invcgn 28195 RingOpscrngo 35053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-1st 7678 df-2nd 7679 df-grpo 28197 df-gid 28198 df-ginv 28199 df-ablo 28249 df-ass 35002 df-exid 35004 df-mgmOLD 35008 df-sgrOLD 35020 df-mndo 35026 df-rngo 35054 |
This theorem is referenced by: rngosubdir 35105 |
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