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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 5948 | . . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
| 4 | 1, 3 | eqtri 2765 | . . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅) |
| 5 | ringnegmul.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 6 | eqid 2737 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
| 7 | 4, 5, 6 | rngo1cl 37946 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋) |
| 8 | ringnegmul.4 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
| 9 | 2, 1, 8 | rngonegcl 37934 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
| 10 | 7, 9 | mpdan 687 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋) |
| 11 | 2, 5, 1 | rngoass 37913 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))) |
| 12 | 11 | 3exp2 1355 | . . . 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 37948 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
| 16 | 15 | 3adant3 1133 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴)) |
| 17 | 16 | oveq1d 7446 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵)) |
| 18 | 2, 5, 1 | rngocl 37908 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
| 19 | 18 | 3expb 1121 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋) |
| 20 | 2, 5, 1, 8, 6 | rngonegmn1l 37948 | . . . 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 2788 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 GIdcgi 30509 invcgn 30510 RingOpscrngo 37901 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902 |
| This theorem is referenced by: rngosubdir 37953 |
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