Theorem rngo1cl 39139

Description: Obsolete theorem, use ringidcl 20545 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 39135	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3	1	eleq1i 2855	. . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp)
4		mndoismgmOLD 39070	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma)
5		mndoisexid 39069	. . . . . . 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 4096	. . . 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 2763	. . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅)
12		ring1cl.3	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
13	1	fveq2i 6919	. . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅))
14	12, 13	eqtri 2786	. . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅))
15	11, 14	iorlid 39058	. . 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 2763	. . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
19		eqid 2763	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
20	18, 19	rngorn1eq 39134	. . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅))
21		eqtr 2783	. . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅))
22	21	eleq2d 2850	. . 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 2147   ∩ cin 4092  ran crn 5659  ‘cfv 6566  1^st c1st 8033  2^nd c2nd 8034  GIdcgi 31149   ExId cexid 39044  Magmacmagm 39048  MndOpcmndo 39066  RingOpscrngo 39094

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-riota 7408  df-ov 7454  df-1st 8035  df-2nd 8036  df-grpo 31152  df-gid 31153  df-ablo 31204  df-ass 39043  df-exid 39045  df-mgmOLD 39049  df-sgrOLD 39061  df-mndo 39067  df-rngo 39095

This theorem is used by:  rngoueqz  39140  rngonegmn1l  39141  rngonegmn1r  39142  rngoneglmul  39143  rngonegrmul  39144  isdrngo2  39158  rngohomco  39174  rngoisocnv  39181  idlnegcl  39222  1idl  39226  0rngo  39227  smprngopr  39252  prnc  39267  isfldidl  39268  isdmn3  39274