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Mirrors > Home > ILE Home > Th. List > dvrcan1 | GIF version |
Description: A cancellation law for division. (divcanap1 8636 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrass.b | ⊢ 𝐵 = (Base‘𝑅) |
dvrass.o | ⊢ 𝑈 = (Unit‘𝑅) |
dvrass.d | ⊢ / = (/r‘𝑅) |
dvrass.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvrcan1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) · 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrass.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝐵 = (Base‘𝑅)) |
3 | dvrass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
4 | 3 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → · = (.r‘𝑅)) |
5 | dvrass.o | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
6 | 5 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑈 = (Unit‘𝑅)) |
7 | eqid 2177 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
8 | 7 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (invr‘𝑅) = (invr‘𝑅)) |
9 | dvrass.d | . . . . 5 ⊢ / = (/r‘𝑅) | |
10 | 9 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → / = (/r‘𝑅)) |
11 | simp1 997 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring) | |
12 | simp2 998 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐵) | |
13 | simp3 999 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
14 | 2, 4, 6, 8, 10, 11, 12, 13 | dvrvald 13256 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · ((invr‘𝑅)‘𝑌))) |
15 | 14 | oveq1d 5889 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) · 𝑌) = ((𝑋 · ((invr‘𝑅)‘𝑌)) · 𝑌)) |
16 | 5, 7, 1 | ringinvcl 13247 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) ∈ 𝐵) |
17 | 16 | 3adant2 1016 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) ∈ 𝐵) |
18 | ringsrg 13177 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
19 | 11, 18 | syl 14 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ SRing) |
20 | 2, 6, 19, 13 | unitcld 13230 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝐵) |
21 | 1, 3 | ringass 13152 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ ((invr‘𝑅)‘𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 · ((invr‘𝑅)‘𝑌)) · 𝑌) = (𝑋 · (((invr‘𝑅)‘𝑌) · 𝑌))) |
22 | 11, 12, 17, 20, 21 | syl13anc 1240 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · ((invr‘𝑅)‘𝑌)) · 𝑌) = (𝑋 · (((invr‘𝑅)‘𝑌) · 𝑌))) |
23 | eqid 2177 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
24 | 5, 7, 3, 23 | unitlinv 13248 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (((invr‘𝑅)‘𝑌) · 𝑌) = (1r‘𝑅)) |
25 | 24 | 3adant2 1016 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (((invr‘𝑅)‘𝑌) · 𝑌) = (1r‘𝑅)) |
26 | 25 | oveq2d 5890 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 · (((invr‘𝑅)‘𝑌) · 𝑌)) = (𝑋 · (1r‘𝑅))) |
27 | 1, 3, 23 | ringridm 13160 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · (1r‘𝑅)) = 𝑋) |
28 | 27 | 3adant3 1017 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 · (1r‘𝑅)) = 𝑋) |
29 | 26, 28 | eqtrd 2210 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 · (((invr‘𝑅)‘𝑌) · 𝑌)) = 𝑋) |
30 | 15, 22, 29 | 3eqtrd 2214 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) · 𝑌) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 .rcmulr 12531 1rcur 13095 SRingcsrg 13099 Ringcrg 13132 Unitcui 13209 invrcinvr 13242 /rcdvr 13253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-tpos 6245 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-cmn 13043 df-abl 13044 df-mgp 13084 df-ur 13096 df-srg 13100 df-ring 13134 df-oppr 13193 df-dvdsr 13211 df-unit 13212 df-invr 13243 df-dvr 13254 |
This theorem is referenced by: dvreq1 13264 lringuplu 13290 |
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