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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 20805 | . . . . . 6 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing)) | |
3 | 1, 2 | sylib 218 | . . . . 5 ⊢ (𝜑 → (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing)) |
4 | 3 | simp3d 1143 | . . . 4 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ DivRing) |
5 | 4 | drngringd 20753 | . . 3 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ Ring) |
6 | sdrgdvcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
7 | 3 | simp2d 1142 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) |
8 | eqid 2734 | . . . . . 6 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
9 | 8 | subrgbas 20597 | . . . . 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 2840 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴))) |
12 | sdrgdvcl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
13 | 12, 10 | eleqtrd 2840 | . . . 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 20595 | . . . . . 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 3007 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴))) |
19 | eqid 2734 | . . . . . 6 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴)) | |
20 | eqid 2734 | . . . . . 6 ⊢ (Unit‘(𝑅 ↾s 𝐴)) = (Unit‘(𝑅 ↾s 𝐴)) | |
21 | eqid 2734 | . . . . . 6 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴)) | |
22 | 19, 20, 21 | drngunit 20750 | . . . . 5 ⊢ ((𝑅 ↾s 𝐴) ∈ DivRing → (𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴)) ↔ (𝑌 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴))))) |
23 | 22 | biimpar 477 | . . . 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 837 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Unit‘(𝑅 ↾s 𝐴))) |
25 | eqid 2734 | . . . 4 ⊢ (/r‘(𝑅 ↾s 𝐴)) = (/r‘(𝑅 ↾s 𝐴)) | |
26 | 19, 20, 25 | dvrcl 20420 | . . 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 1370 | . 2 ⊢ (𝜑 → (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
28 | sdrgdvcl.i | . . . 4 ⊢ / = (/r‘𝑅) | |
29 | 8, 28, 20, 25 | subrgdv 20605 | . . 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 1370 | . 2 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌)) |
31 | 27, 30, 10 | 3eltr4d 2853 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 ↾s cress 17273 0gc0g 17485 Ringcrg 20250 Unitcui 20371 /rcdvr 20416 SubRingcsubrg 20585 DivRingcdr 20745 SubDRingcsdrg 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-subrg 20586 df-drng 20747 df-sdrg 20804 |
This theorem is referenced by: 1fldgenq 33303 constrelextdg2 33751 |
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