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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngolz | Structured version Visualization version GIF version |
Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringlz.1 | ⊢ 𝑍 = (GId‘𝐺) |
ringlz.2 | ⊢ 𝑋 = ran 𝐺 |
ringlz.3 | ⊢ 𝐺 = (1st ‘𝑅) |
ringlz.4 | ⊢ 𝐻 = (2nd ‘𝑅) |
Ref | Expression |
---|---|
rngolz | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringlz.3 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 34250 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringlz.2 | . . . . . . . 8 ⊢ 𝑋 = ran 𝐺 | |
4 | ringlz.1 | . . . . . . . 8 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grpoidcl 27920 | . . . . . . 7 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋) |
6 | 3, 4 | grpolid 27922 | . . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
7 | 5, 6 | mpdan 678 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → (𝑍𝐺𝑍) = 𝑍) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍) |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
10 | 9 | oveq1d 6925 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴)) |
11 | 1, 3, 4 | rngo0cl 34259 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
12 | 11 | adantr 474 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
13 | simpr 479 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
14 | 12, 12, 13 | 3jca 1162 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
15 | ringlz.4 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
16 | 1, 15, 3 | rngodir 34245 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴))) |
17 | 14, 16 | syldan 585 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴))) |
18 | 2 | adantr 474 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp) |
19 | simpl 476 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑅 ∈ RingOps) | |
20 | 1, 15, 3 | rngocl 34241 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋) |
21 | 19, 12, 13, 20 | syl3anc 1494 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋) |
22 | 3, 4 | grporid 27923 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴)) |
23 | 22 | eqcomd 2831 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍)) |
24 | 18, 21, 23 | syl2anc 579 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍)) |
25 | 10, 17, 24 | 3eqtr3d 2869 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍)) |
26 | 3 | grpolcan 27936 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍)) |
27 | 18, 21, 12, 21, 26 | syl13anc 1495 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍)) |
28 | 25, 27 | mpbid 224 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ran crn 5347 ‘cfv 6127 (class class class)co 6910 1st c1st 7431 2nd c2nd 7432 GrpOpcgr 27895 GIdcgi 27896 RingOpscrngo 34234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-1st 7433 df-2nd 7434 df-grpo 27899 df-gid 27900 df-ginv 27901 df-ablo 27951 df-rngo 34235 |
This theorem is referenced by: rngonegmn1l 34281 isdrngo3 34299 0idl 34365 keridl 34372 prnc 34407 |
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