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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 2725 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
3 | subrgdv.3 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑆) | |
4 | eqid 2725 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
5 | 1, 2, 3, 4 | subrginv 20539 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
6 | 5 | 3adant2 1128 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
7 | 6 | oveq2d 7435 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌))) |
8 | eqid 2725 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 1, 8 | ressmulr 17291 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
10 | 9 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆)) |
11 | 10 | oveqd 7436 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
12 | 7, 11 | eqtrd 2765 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
13 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13 | subrgss 20523 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
15 | 14 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 ⊆ (Base‘𝑅)) |
16 | simp2 1134 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐴) | |
17 | 15, 16 | sseldd 3977 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅)) |
18 | eqid 2725 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
19 | 1, 18, 3 | subrguss 20538 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅)) |
20 | 19 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑈 ⊆ (Unit‘𝑅)) |
21 | simp3 1135 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
22 | 20, 21 | sseldd 3977 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Unit‘𝑅)) |
23 | subrgdv.2 | . . . 4 ⊢ / = (/r‘𝑅) | |
24 | 13, 8, 18, 2, 23 | dvrval 20354 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Unit‘𝑅)) → (𝑋 / 𝑌) = (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌))) |
25 | 17, 22, 24 | syl2anc 582 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌))) |
26 | 1 | subrgbas 20532 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
27 | 26 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 = (Base‘𝑆)) |
28 | 16, 27 | eleqtrd 2827 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆)) |
29 | eqid 2725 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
30 | eqid 2725 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
31 | subrgdv.4 | . . . 4 ⊢ 𝐸 = (/r‘𝑆) | |
32 | 29, 30, 3, 4, 31 | dvrval 20354 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ 𝑌 ∈ 𝑈) → (𝑋𝐸𝑌) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
33 | 28, 21, 32 | syl2anc 582 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋𝐸𝑌) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
34 | 12, 25, 33 | 3eqtr4d 2775 | 1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 ↾s cress 17212 .rcmulr 17237 Unitcui 20306 invrcinvr 20338 /rcdvr 20351 SubRingcsubrg 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-subrg 20520 |
This theorem is referenced by: qsssubdrg 21376 redvr 21566 cvsdiv 25103 qrngdiv 27602 sdrgdvcl 33085 |
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