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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 35213 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) |
5 | 4 | simpld 497 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ran crn 5555 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 GIdcgi 28266 RingOpscrngo 35171 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 df-fv 6362 df-riota 7113 df-ov 7158 df-1st 7688 df-2nd 7689 df-grpo 28269 df-gid 28270 df-ablo 28321 df-ass 35120 df-exid 35122 df-mgmOLD 35126 df-sgrOLD 35138 df-mndo 35144 df-rngo 35172 |
This theorem is referenced by: rngonegmn1l 35218 zerdivemp1x 35224 isdrngo2 35235 1idl 35303 smprngopr 35329 prnc 35344 |
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