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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmncan2 | Structured version Visualization version GIF version |
Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
dmncan.1 | ⊢ 𝐺 = (1st ‘𝑅) |
dmncan.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
dmncan.3 | ⊢ 𝑋 = ran 𝐺 |
dmncan.4 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
dmncan2 | ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmncrng 35494 | . . . 4 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) | |
2 | dmncan.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | dmncan.2 | . . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅) | |
4 | dmncan.3 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
5 | 2, 3, 4 | crngocom 35439 | . . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴)) |
6 | 5 | 3adant3r2 1180 | . . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴)) |
7 | 2, 3, 4 | crngocom 35439 | . . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵)) |
8 | 7 | 3adant3r1 1179 | . . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵)) |
9 | 6, 8 | eqeq12d 2814 | . . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
10 | 1, 9 | sylan 583 | . . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
11 | 10 | adantr 484 | . 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
12 | 3anrot 1097 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) | |
13 | 12 | biimpri 231 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) |
14 | dmncan.4 | . . . 4 ⊢ 𝑍 = (GId‘𝐺) | |
15 | 2, 3, 4, 14 | dmncan1 35514 | . . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵)) |
16 | 13, 15 | sylanl2 680 | . 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵)) |
17 | 11, 16 | sylbid 243 | 1 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ran crn 5520 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 GIdcgi 28273 CRingOpsccring 35431 Dmncdmn 35485 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-ass 35281 df-exid 35283 df-mgmOLD 35287 df-sgrOLD 35299 df-mndo 35305 df-rngo 35333 df-com2 35428 df-crngo 35432 df-idl 35448 df-pridl 35449 df-prrngo 35486 df-dmn 35487 df-igen 35498 |
This theorem is referenced by: (None) |
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