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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 20789 | . . . . . 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 1145 | . . . 4 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ DivRing) |
| 5 | 4 | drngringd 20737 | . . 3 ⊢ (𝜑 → (𝑅 ↾s 𝐴) ∈ Ring) |
| 6 | sdrgdvcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
| 7 | 3 | simp2d 1144 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) |
| 8 | eqid 2737 | . . . . . 6 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
| 9 | 8 | subrgbas 20581 | . . . . 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 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴))) |
| 12 | sdrgdvcl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
| 13 | 12, 10 | eleqtrd 2843 | . . . 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 20579 | . . . . . 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 3010 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ (0g‘(𝑅 ↾s 𝐴))) |
| 19 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴)) | |
| 20 | eqid 2737 | . . . . . 6 ⊢ (Unit‘(𝑅 ↾s 𝐴)) = (Unit‘(𝑅 ↾s 𝐴)) | |
| 21 | eqid 2737 | . . . . . 6 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴)) | |
| 22 | 19, 20, 21 | drngunit 20734 | . . . . 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 2737 | . . . 4 ⊢ (/r‘(𝑅 ↾s 𝐴)) = (/r‘(𝑅 ↾s 𝐴)) | |
| 26 | 19, 20, 25 | dvrcl 20404 | . . 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 1373 | . 2 ⊢ (𝜑 → (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
| 28 | sdrgdvcl.i | . . . 4 ⊢ / = (/r‘𝑅) | |
| 29 | 8, 28, 20, 25 | subrgdv 20589 | . . 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 1373 | . 2 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋(/r‘(𝑅 ↾s 𝐴))𝑌)) |
| 31 | 27, 30, 10 | 3eltr4d 2856 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Ringcrg 20230 Unitcui 20355 /rcdvr 20400 SubRingcsubrg 20569 DivRingcdr 20729 SubDRingcsdrg 20787 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrg 20570 df-drng 20731 df-sdrg 20788 |
| This theorem is referenced by: 1fldgenq 33324 constrelextdg2 33788 |
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