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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sdrgdvcl | Structured version Visualization version GIF version |
Description: A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
Ref | Expression |
---|---|
sdrgdvcl.i | ⊢ / = (/r‘𝑅) |
sdrgdvcl.0 | ⊢ 0 = (0g‘𝑅) |
sdrgdvcl.a | ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) |
sdrgdvcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
sdrgdvcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
sdrgdvcl.1 | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
Ref | Expression |
---|---|
sdrgdvcl | ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdrgdvcl.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) | |
2 | issdrg 20392 | . . . . . 6 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing)) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing)) |
4 | 3 | simp3d 1145 | . . . 4 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ DivRing) |
5 | 4 | drngringd 20312 | . . 3 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ Ring) |
6 | sdrgdvcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
7 | 3 | simp2d 1144 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) |
8 | eqid 2733 | . . . . . 6 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
9 | 8 | subrgbas 20360 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
11 | 6, 10 | eleqtrd 2836 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴))) |
12 | sdrgdvcl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
13 | 12, 10 | eleqtrd 2836 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘(𝑅 ↾s 𝐴))) |
14 | sdrgdvcl.1 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
15 | sdrgdvcl.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
16 | 8, 15 | subrg0 20358 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘(𝑅 ↾s 𝐴))) |
17 | 7, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 = (0g‘(𝑅 ↾s 𝐴))) |
18 | 14, 17 | neeqtrd 3011 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴))) |
19 | eqid 2733 | . . . . . 6 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴)) | |
20 | eqid 2733 | . . . . . 6 ⊢ (Unit‘(𝑅 ↾s 𝐴)) = (Unit‘(𝑅 ↾s 𝐴)) | |
21 | eqid 2733 | . . . . . 6 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴)) | |
22 | 19, 20, 21 | drngunit 20309 | . . . . 5 ⊢ ((𝑅 ↾s 𝐴) ∈ DivRing → (𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴)) ↔ (𝑌 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴))))) |
23 | 22 | biimpar 479 | . . . 4 ⊢ (((𝑅 ↾s 𝐴) ∈ DivRing ∧ (𝑌 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴)))) → 𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴))) |
24 | 4, 13, 18, 23 | syl12anc 836 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴))) |
25 | eqid 2733 | . . . 4 ⊢ (/r‘(𝑅 ↾s 𝐴)) = (/r‘(𝑅 ↾s 𝐴)) | |
26 | 19, 20, 25 | dvrcl 20207 | . . 3 ⊢ (((𝑅 ↾s 𝐴) ∈ Ring ∧ 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴))) → (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
27 | 5, 11, 24, 26 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
28 | sdrgdvcl.i | . . . 4 ⊢ / = (/r‘𝑅) | |
29 | 8, 28, 20, 25 | subrgdv 20368 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴))) → (𝑋 / 𝑌) = (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌)) |
30 | 7, 6, 24, 29 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌)) |
31 | 27, 30, 10 | 3eltr4d 2849 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 ↾s cress 17169 0gc0g 17381 Ringcrg 20047 Unitcui 20158 /rcdvr 20203 DivRingcdr 20304 SubRingcsubrg 20347 SubDRingcsdrg 20390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-subrg 20349 df-sdrg 20391 |
This theorem is referenced by: 1fldgenq 32381 |
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