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Mirrors > Home > MPE Home > Th. List > Mathboxes > drngmulcan2ad | Structured version Visualization version GIF version |
Description: Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11846 analog). (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
drngmulcanad.b | ⊢ 𝐵 = (Base‘𝑅) |
drngmulcanad.0 | ⊢ 0 = (0g‘𝑅) |
drngmulcanad.t | ⊢ · = (.r‘𝑅) |
drngmulcanad.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
drngmulcanad.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
drngmulcanad.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
drngmulcanad.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
drngmulcanad.1 | ⊢ (𝜑 → 𝑍 ≠ 0 ) |
drngmulcan2ad.2 | ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍)) |
Ref | Expression |
---|---|
drngmulcan2ad | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngmulcan2ad.2 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍)) | |
2 | 1 | oveq1d 7416 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑍) · ((invr‘𝑅)‘𝑍)) = ((𝑌 · 𝑍) · ((invr‘𝑅)‘𝑍))) |
3 | drngmulcanad.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | drngmulcanad.t | . . . 4 ⊢ · = (.r‘𝑅) | |
5 | drngmulcanad.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
6 | 5 | drngringd 20584 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
7 | drngmulcanad.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | drngmulcanad.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
9 | drngmulcanad.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
10 | eqid 2724 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
11 | drngmulcanad.1 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ 0 ) | |
12 | 3, 9, 10, 5, 8, 11 | drnginvrcld 20600 | . . . 4 ⊢ (𝜑 → ((invr‘𝑅)‘𝑍) ∈ 𝐵) |
13 | 3, 4, 6, 7, 8, 12 | ringassd 20150 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑍) · ((invr‘𝑅)‘𝑍)) = (𝑋 · (𝑍 · ((invr‘𝑅)‘𝑍)))) |
14 | eqid 2724 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
15 | 3, 9, 4, 14, 10, 5, 8, 11 | drnginvrrd 20604 | . . . 4 ⊢ (𝜑 → (𝑍 · ((invr‘𝑅)‘𝑍)) = (1r‘𝑅)) |
16 | 15 | oveq2d 7417 | . . 3 ⊢ (𝜑 → (𝑋 · (𝑍 · ((invr‘𝑅)‘𝑍))) = (𝑋 · (1r‘𝑅))) |
17 | 3, 4, 14, 6, 7 | ringridmd 20161 | . . 3 ⊢ (𝜑 → (𝑋 · (1r‘𝑅)) = 𝑋) |
18 | 13, 16, 17 | 3eqtrd 2768 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑍) · ((invr‘𝑅)‘𝑍)) = 𝑋) |
19 | drngmulcanad.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | 3, 4, 6, 19, 8, 12 | ringassd 20150 | . . 3 ⊢ (𝜑 → ((𝑌 · 𝑍) · ((invr‘𝑅)‘𝑍)) = (𝑌 · (𝑍 · ((invr‘𝑅)‘𝑍)))) |
21 | 15 | oveq2d 7417 | . . 3 ⊢ (𝜑 → (𝑌 · (𝑍 · ((invr‘𝑅)‘𝑍))) = (𝑌 · (1r‘𝑅))) |
22 | 3, 4, 14, 6, 19 | ringridmd 20161 | . . 3 ⊢ (𝜑 → (𝑌 · (1r‘𝑅)) = 𝑌) |
23 | 20, 21, 22 | 3eqtrd 2768 | . 2 ⊢ (𝜑 → ((𝑌 · 𝑍) · ((invr‘𝑅)‘𝑍)) = 𝑌) |
24 | 2, 18, 23 | 3eqtr3d 2772 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 .rcmulr 17196 0gc0g 17383 1rcur 20075 invrcinvr 20278 DivRingcdr 20576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-0g 17385 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-grp 18855 df-minusg 18856 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-drng 20578 |
This theorem is referenced by: drnginvmuld 41558 |
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