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Mirrors > Home > MPE Home > Th. List > subrgdv | Structured version Visualization version GIF version |
Description: A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
subrgdv.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
subrgdv.2 | ⊢ / = (/r‘𝑅) |
subrgdv.3 | ⊢ 𝑈 = (Unit‘𝑆) |
subrgdv.4 | ⊢ 𝐸 = (/r‘𝑆) |
Ref | Expression |
---|---|
subrgdv | ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgdv.1 | . . . . . 6 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | eqid 2818 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
3 | subrgdv.3 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑆) | |
4 | eqid 2818 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
5 | 1, 2, 3, 4 | subrginv 19480 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
6 | 5 | 3adant2 1123 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
7 | 6 | oveq2d 7161 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌))) |
8 | eqid 2818 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 1, 8 | ressmulr 16613 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
10 | 9 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆)) |
11 | 10 | oveqd 7162 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
12 | 7, 11 | eqtrd 2853 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
13 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13 | subrgss 19465 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
15 | 14 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 ⊆ (Base‘𝑅)) |
16 | simp2 1129 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐴) | |
17 | 15, 16 | sseldd 3965 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅)) |
18 | eqid 2818 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
19 | 1, 18, 3 | subrguss 19479 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅)) |
20 | 19 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑈 ⊆ (Unit‘𝑅)) |
21 | simp3 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
22 | 20, 21 | sseldd 3965 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Unit‘𝑅)) |
23 | subrgdv.2 | . . . 4 ⊢ / = (/r‘𝑅) | |
24 | 13, 8, 18, 2, 23 | dvrval 19364 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Unit‘𝑅)) → (𝑋 / 𝑌) = (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌))) |
25 | 17, 22, 24 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌))) |
26 | 1 | subrgbas 19473 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
27 | 26 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 = (Base‘𝑆)) |
28 | 16, 27 | eleqtrd 2912 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆)) |
29 | eqid 2818 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
30 | eqid 2818 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
31 | subrgdv.4 | . . . 4 ⊢ 𝐸 = (/r‘𝑆) | |
32 | 29, 30, 3, 4, 31 | dvrval 19364 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ 𝑌 ∈ 𝑈) → (𝑋𝐸𝑌) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
33 | 28, 21, 32 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋𝐸𝑌) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
34 | 12, 25, 33 | 3eqtr4d 2863 | 1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 .rcmulr 16554 Unitcui 19318 invrcinvr 19350 /rcdvr 19361 SubRingcsubrg 19460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-subrg 19462 |
This theorem is referenced by: qsssubdrg 20532 redvr 20689 cvsdiv 23663 qrngdiv 26127 |
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