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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 2735 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
3 | subrgdv.3 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑆) | |
4 | eqid 2735 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
5 | 1, 2, 3, 4 | subrginv 20605 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
6 | 5 | 3adant2 1130 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌)) |
7 | 6 | oveq2d 7447 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌))) |
8 | eqid 2735 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 1, 8 | ressmulr 17353 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
10 | 9 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆)) |
11 | 10 | oveqd 7448 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
12 | 7, 11 | eqtrd 2775 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌))) |
13 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13 | subrgss 20589 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
15 | 14 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 ⊆ (Base‘𝑅)) |
16 | simp2 1136 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐴) | |
17 | 15, 16 | sseldd 3996 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅)) |
18 | eqid 2735 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
19 | 1, 18, 3 | subrguss 20604 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅)) |
20 | 19 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑈 ⊆ (Unit‘𝑅)) |
21 | simp3 1137 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
22 | 20, 21 | sseldd 3996 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Unit‘𝑅)) |
23 | subrgdv.2 | . . . 4 ⊢ / = (/r‘𝑅) | |
24 | 13, 8, 18, 2, 23 | dvrval 20420 | . . 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 20598 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
27 | 26 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 = (Base‘𝑆)) |
28 | 16, 27 | eleqtrd 2841 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆)) |
29 | eqid 2735 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
30 | eqid 2735 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
31 | subrgdv.4 | . . . 4 ⊢ 𝐸 = (/r‘𝑆) | |
32 | 29, 30, 3, 4, 31 | dvrval 20420 | . . 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 2785 | 1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Unitcui 20372 invrcinvr 20404 /rcdvr 20417 SubRingcsubrg 20586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrg 20587 |
This theorem is referenced by: qsssubdrg 21462 redvr 21653 cvsdiv 25179 qrngdiv 27683 sdrgdvcl 33281 |
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