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Mirrors > Home > MPE Home > Th. List > dvrcan3 | Structured version Visualization version GIF version |
Description: A cancellation law for division. (divcan3 11885 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
dvrass.b | ⊢ 𝐵 = (Base‘𝑅) |
dvrass.o | ⊢ 𝑈 = (Unit‘𝑅) |
dvrass.d | ⊢ / = (/r‘𝑅) |
dvrass.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvrcan3 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1137 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring) | |
2 | simp2 1138 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐵) | |
3 | dvrass.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | dvrass.o | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 20167 | . . . 4 ⊢ (𝑌 ∈ 𝑈 → 𝑌 ∈ 𝐵) |
6 | 5 | 3ad2ant3 1136 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝐵) |
7 | simp3 1139 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
8 | dvrass.d | . . . 4 ⊢ / = (/r‘𝑅) | |
9 | dvrass.t | . . . 4 ⊢ · = (.r‘𝑅) | |
10 | 3, 4, 8, 9 | dvrass 20200 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑌) = (𝑋 · (𝑌 / 𝑌))) |
11 | 1, 2, 6, 7, 10 | syl13anc 1373 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑌) = (𝑋 · (𝑌 / 𝑌))) |
12 | eqid 2733 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
13 | 4, 8, 12 | dvrid 20198 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑌 / 𝑌) = (1r‘𝑅)) |
14 | 13 | 3adant2 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑌 / 𝑌) = (1r‘𝑅)) |
15 | 14 | oveq2d 7412 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 · (𝑌 / 𝑌)) = (𝑋 · (1r‘𝑅))) |
16 | 3, 9, 12 | ringridm 20068 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · (1r‘𝑅)) = 𝑋) |
17 | 16 | 3adant3 1133 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 · (1r‘𝑅)) = 𝑋) |
18 | 11, 15, 17 | 3eqtrd 2777 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 .rcmulr 17185 1rcur 19987 Ringcrg 20038 Unitcui 20147 /rcdvr 20192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-0g 17374 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 |
This theorem is referenced by: irredrmul 20219 cramerimp 22157 lgseisenlem3 26847 orngrmullt 32388 |
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