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Mirrors > Home > MPE Home > Th. List > Mathboxes > domnlcanbOLD | Structured version Visualization version GIF version |
Description: Obsolete version of domnlcanb 20742 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
domncanOLD.b | ⊢ 𝐵 = (Base‘𝑅) |
domncanOLD.1 | ⊢ 0 = (0g‘𝑅) |
domncanOLD.m | ⊢ · = (.r‘𝑅) |
domncanOLD.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) |
domncanOLD.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
domncanOLD.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
domnlcanbOLD.r | ⊢ (𝜑 → 𝑅 ∈ Domn) |
Ref | Expression |
---|---|
domnlcanbOLD | ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domncanOLD.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | domncanOLD.1 | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | domncanOLD.m | . . 3 ⊢ · = (.r‘𝑅) | |
4 | domncanOLD.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → 𝑋 ∈ (𝐵 ∖ { 0 })) |
6 | domncanOLD.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → 𝑌 ∈ 𝐵) |
8 | domncanOLD.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → 𝑍 ∈ 𝐵) |
10 | domnlcanbOLD.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → 𝑅 ∈ Domn) |
12 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → (𝑋 · 𝑌) = (𝑋 · 𝑍)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | domnlcan 20743 | . 2 ⊢ ((𝜑 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑍)) → 𝑌 = 𝑍) |
14 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍) | |
15 | 14 | oveq2d 7464 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑋 · 𝑌) = (𝑋 · 𝑍)) |
16 | 13, 15 | impbida 800 | 1 ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 {csn 4648 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 .rcmulr 17312 0gc0g 17499 Domncdomn 20714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-nzr 20539 df-domn 20717 |
This theorem is referenced by: (None) |
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