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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 | β’ (π β π΅ = (Baseβπ )) |
dvrvald.t | β’ (π β Β· = (.rβπ )) |
dvrvald.u | β’ (π β π = (Unitβπ )) |
dvrvald.i | β’ (π β πΌ = (invrβπ )) |
dvrvald.d | β’ (π β / = (/rβπ )) |
dvrvald.r | β’ (π β π β Ring) |
dvrvald.x | β’ (π β π β π΅) |
dvrvald.y | β’ (π β π β π) |
Ref | Expression |
---|---|
dvrvald | β’ (π β (π / π) = (π Β· (πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrvald.b | . . 3 β’ (π β π΅ = (Baseβπ )) | |
2 | dvrvald.t | . . 3 β’ (π β Β· = (.rβπ )) | |
3 | dvrvald.u | . . 3 β’ (π β π = (Unitβπ )) | |
4 | dvrvald.i | . . 3 β’ (π β πΌ = (invrβπ )) | |
5 | dvrvald.d | . . 3 β’ (π β / = (/rβπ )) | |
6 | dvrvald.r | . . . 4 β’ (π β π β Ring) | |
7 | ringsrg 13297 | . . . 4 β’ (π β Ring β π β SRing) | |
8 | 6, 7 | syl 14 | . . 3 β’ (π β π β SRing) |
9 | 1, 2, 3, 4, 5, 8 | dvrfvald 13381 | . 2 β’ (π β / = (π₯ β π΅, π¦ β π β¦ (π₯ Β· (πΌβπ¦)))) |
10 | simpl 109 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β π₯ = π) | |
11 | fveq2 5527 | . . . . 5 β’ (π¦ = π β (πΌβπ¦) = (πΌβπ)) | |
12 | 11 | adantl 277 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β (πΌβπ¦) = (πΌβπ)) |
13 | 10, 12 | oveq12d 5906 | . . 3 β’ ((π₯ = π β§ π¦ = π) β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ))) |
14 | 13 | adantl 277 | . 2 β’ ((π β§ (π₯ = π β§ π¦ = π)) β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ))) |
15 | dvrvald.x | . 2 β’ (π β π β π΅) | |
16 | dvrvald.y | . 2 β’ (π β π β π) | |
17 | 2 | oveqd 5905 | . . 3 β’ (π β (π Β· (πΌβπ)) = (π(.rβπ )(πΌβπ))) |
18 | 15, 1 | eleqtrd 2266 | . . . 4 β’ (π β π β (Baseβπ )) |
19 | eqidd 2188 | . . . . 5 β’ (π β (Baseβπ ) = (Baseβπ )) | |
20 | 16, 3 | eleqtrd 2266 | . . . . . . 7 β’ (π β π β (Unitβπ )) |
21 | eqid 2187 | . . . . . . . 8 β’ (Unitβπ ) = (Unitβπ ) | |
22 | eqid 2187 | . . . . . . . 8 β’ (invrβπ ) = (invrβπ ) | |
23 | 21, 22 | unitinvcl 13371 | . . . . . . 7 β’ ((π β Ring β§ π β (Unitβπ )) β ((invrβπ )βπ) β (Unitβπ )) |
24 | 6, 20, 23 | syl2anc 411 | . . . . . 6 β’ (π β ((invrβπ )βπ) β (Unitβπ )) |
25 | 4 | fveq1d 5529 | . . . . . 6 β’ (π β (πΌβπ) = ((invrβπ )βπ)) |
26 | 24, 25, 3 | 3eltr4d 2271 | . . . . 5 β’ (π β (πΌβπ) β π) |
27 | 19, 3, 8, 26 | unitcld 13356 | . . . 4 β’ (π β (πΌβπ) β (Baseβπ )) |
28 | eqid 2187 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
29 | eqid 2187 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
30 | 28, 29 | ringcl 13265 | . . . 4 β’ ((π β Ring β§ π β (Baseβπ ) β§ (πΌβπ) β (Baseβπ )) β (π(.rβπ )(πΌβπ)) β (Baseβπ )) |
31 | 6, 18, 27, 30 | syl3anc 1248 | . . 3 β’ (π β (π(.rβπ )(πΌβπ)) β (Baseβπ )) |
32 | 17, 31 | eqeltrd 2264 | . 2 β’ (π β (π Β· (πΌβπ)) β (Baseβπ )) |
33 | 9, 14, 15, 16, 32 | ovmpod 6016 | 1 β’ (π β (π / π) = (π Β· (πΌβπ))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2158 βcfv 5228 (class class class)co 5888 Basecbs 12476 .rcmulr 12552 SRingcsrg 13215 Ringcrg 13248 Unitcui 13335 invrcinvr 13368 /rcdvr 13379 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-lttrn 7939 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-tpos 6260 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-inn 8934 df-2 8992 df-3 8993 df-ndx 12479 df-slot 12480 df-base 12482 df-sets 12483 df-iress 12484 df-plusg 12564 df-mulr 12565 df-0g 12725 df-mgm 12794 df-sgrp 12827 df-mnd 12840 df-grp 12902 df-minusg 12903 df-cmn 13123 df-abl 13124 df-mgp 13173 df-ur 13212 df-srg 13216 df-ring 13250 df-oppr 13316 df-dvdsr 13337 df-unit 13338 df-invr 13369 df-dvr 13380 |
This theorem is referenced by: dvrcl 13383 unitdvcl 13384 dvrid 13385 dvr1 13386 dvrass 13387 dvrcan1 13388 dvrdir 13391 rdivmuldivd 13392 ringinvdv 13393 subrgdv 13458 |
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