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Mirrors > Home > ILE Home > Th. List > dvrvald | GIF version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrvald.b | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
dvrvald.t | ⊢ (𝜑 → · = (.r‘𝑅)) |
dvrvald.u | ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) |
dvrvald.i | ⊢ (𝜑 → 𝐼 = (invr‘𝑅)) |
dvrvald.d | ⊢ (𝜑 → / = (/r‘𝑅)) |
dvrvald.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
dvrvald.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
dvrvald.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Ref | Expression |
---|---|
dvrvald | ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrvald.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | dvrvald.t | . . 3 ⊢ (𝜑 → · = (.r‘𝑅)) | |
3 | dvrvald.u | . . 3 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) | |
4 | dvrvald.i | . . 3 ⊢ (𝜑 → 𝐼 = (invr‘𝑅)) | |
5 | dvrvald.d | . . 3 ⊢ (𝜑 → / = (/r‘𝑅)) | |
6 | dvrvald.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | ringsrg 13235 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝜑 → 𝑅 ∈ SRing) |
9 | 1, 2, 3, 4, 5, 8 | dvrfvald 13313 | . 2 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
10 | simpl 109 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
11 | fveq2 5517 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
12 | 11 | adantl 277 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐼‘𝑦) = (𝐼‘𝑌)) |
13 | 10, 12 | oveq12d 5896 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑌))) |
14 | 13 | adantl 277 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑌))) |
15 | dvrvald.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | dvrvald.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
17 | 2 | oveqd 5895 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐼‘𝑌)) = (𝑋(.r‘𝑅)(𝐼‘𝑌))) |
18 | 15, 1 | eleqtrd 2256 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) |
19 | eqidd 2178 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅)) | |
20 | 16, 3 | eleqtrd 2256 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Unit‘𝑅)) |
21 | eqid 2177 | . . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
22 | eqid 2177 | . . . . . . . 8 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
23 | 21, 22 | unitinvcl 13303 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Unit‘𝑅)) → ((invr‘𝑅)‘𝑌) ∈ (Unit‘𝑅)) |
24 | 6, 20, 23 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → ((invr‘𝑅)‘𝑌) ∈ (Unit‘𝑅)) |
25 | 4 | fveq1d 5519 | . . . . . 6 ⊢ (𝜑 → (𝐼‘𝑌) = ((invr‘𝑅)‘𝑌)) |
26 | 24, 25, 3 | 3eltr4d 2261 | . . . . 5 ⊢ (𝜑 → (𝐼‘𝑌) ∈ 𝑈) |
27 | 19, 3, 8, 26 | unitcld 13288 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (Base‘𝑅)) |
28 | eqid 2177 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
29 | eqid 2177 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
30 | 28, 29 | ringcl 13207 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝐼‘𝑌) ∈ (Base‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑌)) ∈ (Base‘𝑅)) |
31 | 6, 18, 27, 30 | syl3anc 1238 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)(𝐼‘𝑌)) ∈ (Base‘𝑅)) |
32 | 17, 31 | eqeltrd 2254 | . 2 ⊢ (𝜑 → (𝑋 · (𝐼‘𝑌)) ∈ (Base‘𝑅)) |
33 | 9, 14, 15, 16, 32 | ovmpod 6005 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5218 (class class class)co 5878 Basecbs 12465 .rcmulr 12540 SRingcsrg 13157 Ringcrg 13190 Unitcui 13267 invrcinvr 13300 /rcdvr 13311 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-tpos 6249 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-3 8982 df-ndx 12468 df-slot 12469 df-base 12471 df-sets 12472 df-iress 12473 df-plusg 12552 df-mulr 12553 df-0g 12713 df-mgm 12782 df-sgrp 12815 df-mnd 12825 df-grp 12887 df-minusg 12888 df-cmn 13101 df-abl 13102 df-mgp 13142 df-ur 13154 df-srg 13158 df-ring 13192 df-oppr 13251 df-dvdsr 13269 df-unit 13270 df-invr 13301 df-dvr 13312 |
This theorem is referenced by: dvrcl 13315 unitdvcl 13316 dvrid 13317 dvr1 13318 dvrass 13319 dvrcan1 13320 dvrdir 13323 rdivmuldivd 13324 ringinvdv 13325 subrgdv 13370 |
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