Theorem rngo1cl 37923

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 37919	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3	1	eleq1i 2819	. . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp)
4		mndoismgmOLD 37854	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma)
5		mndoisexid 37853	. . . . . . 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 3919	. . . 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 2729	. . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅)
12		ring1cl.3	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
13	1	fveq2i 6825	. . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅))
14	12, 13	eqtri 2752	. . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅))
15	11, 14	iorlid 37842	. . 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 2729	. . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
19		eqid 2729	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
20	18, 19	rngorn1eq 37918	. . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅))
21		eqtr 2749	. . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅))
22	21	eleq2d 2814	. . 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 1540   ∈ wcel 2109   ∩ cin 3902  ran crn 5620  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923  GIdcgi 30434   ExId cexid 37828  Magmacmagm 37832  MndOpcmndo 37850  RingOpscrngo 37878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-riota 7306  df-ov 7352  df-1st 7924  df-2nd 7925  df-grpo 30437  df-gid 30438  df-ablo 30489  df-ass 37827  df-exid 37829  df-mgmOLD 37833  df-sgrOLD 37845  df-mndo 37851  df-rngo 37879

This theorem is referenced by:  rngoueqz  37924  rngonegmn1l  37925  rngonegmn1r  37926  rngoneglmul  37927  rngonegrmul  37928  isdrngo2  37942  rngohomco  37958  rngoisocnv  37965  idlnegcl  38006  1idl  38010  0rngo  38011  smprngopr  38036  prnc  38051  isfldidl  38052  isdmn3  38058