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Mirrors > Home > MPE Home > Th. List > domnlcan | Structured version Visualization version GIF version |
Description: Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.) |
Ref | Expression |
---|---|
domncan.b | ⊢ 𝐵 = (Base‘𝑅) |
domncan.0 | ⊢ 0 = (0g‘𝑅) |
domncan.m | ⊢ · = (.r‘𝑅) |
domncan.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) |
domncan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
domncan.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
domncan.r | ⊢ (𝜑 → 𝑅 ∈ Domn) |
domnlcan.1 | ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍)) |
Ref | Expression |
---|---|
domnlcan | ⊢ (𝜑 → 𝑌 = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnlcan.1 | . 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍)) | |
2 | domncan.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | domncan.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | domncan.m | . . 3 ⊢ · = (.r‘𝑅) | |
5 | domncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) | |
6 | domncan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | domncan.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
8 | domncan.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
9 | 2, 3, 4, 5, 6, 7, 8 | domnlcanb 20736 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍)) |
10 | 1, 9 | mpbid 232 | 1 ⊢ (𝜑 → 𝑌 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 {csn 4630 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 .rcmulr 17298 0gc0g 17485 Domncdomn 20708 |
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-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3044 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-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 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-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 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-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-nzr 20529 df-domn 20711 |
This theorem is referenced by: domnlcanbOLD 33264 idomrcanOLD 33265 rprmasso2 33533 1arithufdlem3 33553 drngmullcan 42511 fidomncyc 42521 |
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