Theorem rngo1cl 37926

Description: The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ring1cl.1	⊢ 𝑋 = ran (1st ‘𝑅)
ring1cl.2	⊢ 𝐻 = (2nd ‘𝑅)
ring1cl.3	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
rngo1cl	⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)

Proof of Theorem rngo1cl

Step	Hyp	Ref	Expression
1		ring1cl.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
2	1	rngomndo 37922	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3	1	eleq1i 2830	. . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp)
4		mndoismgmOLD 37857	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma)
5		mndoisexid 37856	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId )
6	4, 5	jca 511	. . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
7	3, 6	sylbi 217	. . . . 5 ⊢ (𝐻 ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
8	2, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ RingOps → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
9		elin 3979	. . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
10	8, 9	sylibr 234	. . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId ))
11		eqid 2735	. . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅)
12		ring1cl.3	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
13	1	fveq2i 6910	. . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅))
14	12, 13	eqtri 2763	. . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅))
15	11, 14	iorlid 37845	. . 3 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) → 𝑈 ∈ ran (2nd ‘𝑅))
16	10, 15	syl 17	. 2 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran (2nd ‘𝑅))
17		ring1cl.1	. . 3 ⊢ 𝑋 = ran (1st ‘𝑅)
18		eqid 2735	. . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
19		eqid 2735	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
20	18, 19	rngorn1eq 37921	. . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅))
21		eqtr 2758	. . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅))
22	21	eleq2d 2825	. . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅)))
23	17, 20, 22	sylancr 587	. 2 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅)))
24	16, 23	mpbird 257	1 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∩ cin 3962  ran crn 5690  ‘cfv 6563  1^st c1st 8011  2^nd c2nd 8012  GIdcgi 30519   ExId cexid 37831  Magmacmagm 37835  MndOpcmndo 37853  RingOpscrngo 37881

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882

This theorem is referenced by:  rngoueqz  37927  rngonegmn1l  37928  rngonegmn1r  37929  rngoneglmul  37930  rngonegrmul  37931  isdrngo2  37945  rngohomco  37961  rngoisocnv  37968  idlnegcl  38009  1idl  38013  0rngo  38014  smprngopr  38039  prnc  38054  isfldidl  38055  isdmn3  38061