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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoidmlem | Structured version Visualization version GIF version |
Description: The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
uridm.1 | ⊢ 𝐻 = (2nd ‘𝑅) |
uridm.2 | ⊢ 𝑋 = ran (1st ‘𝑅) |
uridm.3 | ⊢ 𝑈 = (GId‘𝐻) |
Ref | Expression |
---|---|
rngoidmlem | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uridm.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
2 | 1 | rngomndo 36394 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
3 | mndomgmid 36330 | . . . 4 ⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId )) | |
4 | eqid 2736 | . . . . . 6 ⊢ ran 𝐻 = ran 𝐻 | |
5 | uridm.3 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
6 | 4, 5 | cmpidelt 36318 | . . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
7 | 6 | ex 413 | . . . 4 ⊢ (𝐻 ∈ (Magma ∩ ExId ) → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))) |
8 | 2, 3, 7 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))) |
9 | eqid 2736 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
10 | 1, 9 | rngorn1eq 36393 | . . . 4 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran 𝐻) |
11 | uridm.2 | . . . . 5 ⊢ 𝑋 = ran (1st ‘𝑅) | |
12 | eqtr 2759 | . . . . . 6 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran 𝐻) → 𝑋 = ran 𝐻) | |
13 | simpl 483 | . . . . . . . . 9 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → 𝑋 = ran 𝐻) | |
14 | 13 | eleq2d 2823 | . . . . . . . 8 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ran 𝐻)) |
15 | 14 | imbi1d 341 | . . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))) |
16 | 15 | ex 413 | . . . . . 6 ⊢ (𝑋 = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))) |
17 | 12, 16 | syl 17 | . . . . 5 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran 𝐻) → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))) |
18 | 11, 17 | mpan 688 | . . . 4 ⊢ (ran (1st ‘𝑅) = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))) |
19 | 10, 18 | mpcom 38 | . . 3 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))) |
20 | 8, 19 | mpbird 256 | . 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))) |
21 | 20 | imp 407 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 ran crn 5634 ‘cfv 6496 (class class class)co 7357 1st c1st 7919 2nd c2nd 7920 GIdcgi 29432 ExId cexid 36303 Magmacmagm 36307 MndOpcmndo 36325 RingOpscrngo 36353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fo 6502 df-fv 6504 df-riota 7313 df-ov 7360 df-1st 7921 df-2nd 7922 df-grpo 29435 df-gid 29436 df-ablo 29487 df-ass 36302 df-exid 36304 df-mgmOLD 36308 df-sgrOLD 36320 df-mndo 36326 df-rngo 36354 |
This theorem is referenced by: rngolidm 36396 rngoridm 36397 |
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