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Mirrors > Home > MPE Home > Th. List > dvreq1 | Structured version Visualization version GIF version |
Description: Equality in terms of ratio equal to ring unity. (diveq1 11901 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dvreq1.b | ⊢ 𝐵 = (Base‘𝑅) |
dvreq1.o | ⊢ 𝑈 = (Unit‘𝑅) |
dvreq1.d | ⊢ / = (/r‘𝑅) |
dvreq1.t | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
dvreq1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) = 1 ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7412 | . . 3 ⊢ ((𝑋 / 𝑌) = 1 → ((𝑋 / 𝑌)(.r‘𝑅)𝑌) = ( 1 (.r‘𝑅)𝑌)) | |
2 | dvreq1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
3 | dvreq1.o | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
4 | dvreq1.d | . . . . 5 ⊢ / = (/r‘𝑅) | |
5 | eqid 2732 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | 2, 3, 4, 5 | dvrcan1 20215 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌)(.r‘𝑅)𝑌) = 𝑋) |
7 | 2, 3 | unitcl 20181 | . . . . . 6 ⊢ (𝑌 ∈ 𝑈 → 𝑌 ∈ 𝐵) |
8 | dvreq1.t | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
9 | 2, 5, 8 | ringlidm 20079 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑌) = 𝑌) |
10 | 7, 9 | sylan2 593 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → ( 1 (.r‘𝑅)𝑌) = 𝑌) |
11 | 10 | 3adant2 1131 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ( 1 (.r‘𝑅)𝑌) = 𝑌) |
12 | 6, 11 | eqeq12d 2748 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (((𝑋 / 𝑌)(.r‘𝑅)𝑌) = ( 1 (.r‘𝑅)𝑌) ↔ 𝑋 = 𝑌)) |
13 | 1, 12 | imbitrid 243 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) = 1 → 𝑋 = 𝑌)) |
14 | 3, 4, 8 | dvrid 20212 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑌 / 𝑌) = 1 ) |
15 | 14 | 3adant2 1131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑌 / 𝑌) = 1 ) |
16 | oveq1 7412 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 / 𝑌) = (𝑌 / 𝑌)) | |
17 | 16 | eqeq1d 2734 | . . 3 ⊢ (𝑋 = 𝑌 → ((𝑋 / 𝑌) = 1 ↔ (𝑌 / 𝑌) = 1 )) |
18 | 15, 17 | syl5ibrcom 246 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 = 𝑌 → (𝑋 / 𝑌) = 1 )) |
19 | 13, 18 | impbid 211 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) = 1 ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 .rcmulr 17194 1rcur 19998 Ringcrg 20049 Unitcui 20161 /rcdvr 20206 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 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 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 |
This theorem is referenced by: lringuplu 20306 sum2dchr 26766 |
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