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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoidmlem | Structured version Visualization version GIF version |
Description: The unit 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 35207 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
3 | mndomgmid 35143 | . . . 4 ⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId )) | |
4 | eqid 2821 | . . . . . 6 ⊢ ran 𝐻 = ran 𝐻 | |
5 | uridm.3 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
6 | 4, 5 | cmpidelt 35131 | . . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
7 | 6 | ex 415 | . . . 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 2821 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
10 | 1, 9 | rngorn1eq 35206 | . . . 4 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran 𝐻) |
11 | uridm.2 | . . . . 5 ⊢ 𝑋 = ran (1st ‘𝑅) | |
12 | eqtr 2841 | . . . . . 6 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran 𝐻) → 𝑋 = ran 𝐻) | |
13 | simpl 485 | . . . . . . . . 9 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → 𝑋 = ran 𝐻) | |
14 | 13 | eleq2d 2898 | . . . . . . . 8 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ran 𝐻)) |
15 | 14 | imbi1d 344 | . . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))) |
16 | 15 | ex 415 | . . . . . 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 259 | . 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))) |
21 | 20 | imp 409 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ran crn 5550 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 GIdcgi 28261 ExId cexid 35116 Magmacmagm 35120 MndOpcmndo 35138 RingOpscrngo 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 |
This theorem is referenced by: rngolidm 35209 rngoridm 35210 |
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