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Mathbox for Jeff Madsen |
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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 5891 | . . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
4 | 1, 3 | eqtri 2764 | . . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅) |
5 | ringnegmul.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | eqid 2736 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
7 | 4, 5, 6 | rngo1cl 36387 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋) |
8 | ringnegmul.4 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
9 | 2, 1, 8 | rngonegcl 36375 | . . . . 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 36354 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
12 | 11 | 3exp2 1354 | . . . 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 1111 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
15 | 2, 5, 1, 8, 6 | rngonegmn1l 36389 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
16 | 15 | 3adant3 1132 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
17 | 16 | oveq1d 7369 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵)) |
18 | 2, 5, 1 | rngocl 36349 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
19 | 18 | 3expb 1120 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋) |
20 | 2, 5, 1, 8, 6 | rngonegmn1l 36389 | . . . 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 1115 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
23 | 14, 17, 22 | 3eqtr4rd 2787 | 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 1087 = wceq 1541 ∈ wcel 2106 ran crn 5633 ‘cfv 6494 (class class class)co 7354 1st c1st 7916 2nd c2nd 7917 GIdcgi 29330 invcgn 29331 RingOpscrngo 36342 |
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 5241 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-1st 7918 df-2nd 7919 df-grpo 29333 df-gid 29334 df-ginv 29335 df-ablo 29385 df-ass 36291 df-exid 36293 df-mgmOLD 36297 df-sgrOLD 36309 df-mndo 36315 df-rngo 36343 |
This theorem is referenced by: rngosubdir 36394 |
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