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Mirrors > Home > MPE Home > Th. List > sdrgunit | Structured version Visualization version GIF version |
Description: A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.) |
Ref | Expression |
---|---|
sdrgunit.s | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
sdrgunit.0 | ⊢ 0 = (0g‘𝑅) |
sdrgunit.u | ⊢ 𝑈 = (Unit‘𝑆) |
Ref | Expression |
---|---|
sdrgunit | ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdrgunit.s | . . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | 1 | sdrgdrng 20677 | . . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing) |
3 | eqid 2728 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | sdrgunit.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑆) | |
5 | eqid 2728 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | 3, 4, 5 | drngunit 20628 | . . 3 ⊢ (𝑆 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (Base‘𝑆) ∧ 𝑋 ≠ (0g‘𝑆)))) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (Base‘𝑆) ∧ 𝑋 ≠ (0g‘𝑆)))) |
8 | 1 | sdrgbas 20681 | . . . 4 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
9 | 8 | eleq2d 2815 | . . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (Base‘𝑆))) |
10 | sdrgsubrg 20678 | . . . . 5 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅)) | |
11 | sdrgunit.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
12 | 1, 11 | subrg0 20517 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆)) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 0 = (0g‘𝑆)) |
14 | 13 | neeq2d 2998 | . . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ≠ 0 ↔ 𝑋 ≠ (0g‘𝑆))) |
15 | 9, 14 | anbi12d 631 | . 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → ((𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) ↔ (𝑋 ∈ (Base‘𝑆) ∧ 𝑋 ≠ (0g‘𝑆)))) |
16 | 7, 15 | bitr4d 282 | 1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 ↾s cress 17208 0gc0g 17420 Unitcui 20293 SubRingcsubrg 20505 DivRingcdr 20623 SubDRingcsdrg 20673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-subg 19077 df-ring 20174 df-subrg 20507 df-drng 20625 df-sdrg 20674 |
This theorem is referenced by: imadrhmcl 20684 |
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