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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngolidm | Structured version Visualization version GIF version |
Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-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 |
---|---|
rngolidm | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uridm.1 | . . 3 ⊢ 𝐻 = (2nd ‘𝑅) | |
2 | uridm.2 | . . 3 ⊢ 𝑋 = ran (1st ‘𝑅) | |
3 | uridm.3 | . . 3 ⊢ 𝑈 = (GId‘𝐻) | |
4 | 1, 2, 3 | rngoidmlem 35095 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
5 | 4 | simpld 495 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ran crn 5549 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 RingOpscrngo 35053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7103 df-ov 7148 df-1st 7678 df-2nd 7679 df-grpo 28197 df-gid 28198 df-ablo 28249 df-ass 35002 df-exid 35004 df-mgmOLD 35008 df-sgrOLD 35020 df-mndo 35026 df-rngo 35054 |
This theorem is referenced by: rngonegmn1l 35100 zerdivemp1x 35106 isdrngo2 35117 1idl 35185 smprngopr 35211 prnc 35226 |
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