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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 38042 | . . . 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 37987 | . . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴)) |
6 | 5 | 3adant3r2 1182 | . . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴)) |
7 | 2, 3, 4 | crngocom 37987 | . . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵)) |
8 | 7 | 3adant3r1 1181 | . . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵)) |
9 | 6, 8 | eqeq12d 2750 | . . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
10 | 1, 9 | sylan 580 | . . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
11 | 10 | adantr 480 | . 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵))) |
12 | 3anrot 1099 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) | |
13 | 12 | biimpri 228 | . . 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 38062 | . . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵)) |
16 | 13, 15 | sylanl2 681 | . 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵)) |
17 | 11, 16 | sylbid 240 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ran crn 5689 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 GIdcgi 30518 CRingOpsccring 37979 Dmncdmn 38033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-1o 8504 df-en 8984 df-grpo 30521 df-gid 30522 df-ginv 30523 df-gdiv 30524 df-ablo 30573 df-ass 37829 df-exid 37831 df-mgmOLD 37835 df-sgrOLD 37847 df-mndo 37853 df-rngo 37881 df-com2 37976 df-crngo 37980 df-idl 37996 df-pridl 37997 df-prrngo 38034 df-dmn 38035 df-igen 38046 |
This theorem is referenced by: (None) |
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