Theorem rngo1cl 38650

Description: Obsolete theorem, use ringgrp 20366 instead. The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.) (Proof modification 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 38646	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3	1	eleq1i 2856	. . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp)
4		mndoismgmOLD 38581	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma)
5		mndoisexid 38580	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId )
6	4, 5	jca 521	. . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
7	3, 6	sylbi 220	. . . . 5 ⊢ (𝐻 ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
8	2, 7	syl 18	. . . 4 ⊢ (𝑅 ∈ RingOps → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
9		elin 3922	. . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId ))
10	8, 9	sylibr 237	. . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId ))
11		eqid 2765	. . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅)
12		ring1cl.3	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
13	1	fveq2i 6888	. . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅))
14	12, 13	eqtri 2788	. . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅))
15	11, 14	iorlid 38569	. . 3 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) → 𝑈 ∈ ran (2nd ‘𝑅))
16	10, 15	syl 18	. 2 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran (2nd ‘𝑅))
17		ring1cl.1	. . 3 ⊢ 𝑋 = ran (1st ‘𝑅)
18		eqid 2765	. . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
19		eqid 2765	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
20	18, 19	rngorn1eq 38645	. . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅))
21		eqtr 2785	. . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅))
22	21	eleq2d 2851	. . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅)))
23	17, 20, 22	sylancr 599	. 2 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅)))
24	16, 23	mpbird 260	1 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∩ cin 3905  ran crn 5664  ‘cfv 6540  1^st c1st 7990  2^nd c2nd 7991  GIdcgi 30915   ExId cexid 38555  Magmacmagm 38559  MndOpcmndo 38577  RingOpscrngo 38605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-riota 7376  df-ov 7422  df-1st 7992  df-2nd 7993  df-grpo 30918  df-gid 30919  df-ablo 30970  df-ass 38554  df-exid 38556  df-mgmOLD 38560  df-sgrOLD 38572  df-mndo 38578  df-rngo 38606

This theorem is used by:  rngoueqz  38651  rngonegmn1l  38652  rngonegmn1r  38653  rngoneglmul  38654  rngonegrmul  38655  isdrngo2  38669  rngohomco  38685  rngoisocnv  38692  idlnegcl  38733  1idl  38737  0rngo  38738  smprngopr  38763  prnc  38778  isfldidl  38779  isdmn3  38785