Theorem rngo1cl 39013

Description: Obsolete theorem, use ringidcl 20506 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 39009	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3	1	eleq1i 2854	. . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp)
4		mndoismgmOLD 38944	. . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma)
5		mndoisexid 38943	. . . . . . 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 4095	. . . 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 6896	. . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅))
14	12, 13	eqtri 2786	. . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅))
15	11, 14	iorlid 38932	. . 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 39008	. . 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 2849	. . 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 1569   ∈ wcel 2146   ∩ cin 4091  ran crn 5655  ‘cfv 6546  1^st c1st 8006  2^nd c2nd 8007  GIdcgi 31104   ExId cexid 38918  Magmacmagm 38922  MndOpcmndo 38940  RingOpscrngo 38968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fo 6552  df-fv 6554  df-riota 7384  df-ov 7430  df-1st 8008  df-2nd 8009  df-grpo 31107  df-gid 31108  df-ablo 31159  df-ass 38917  df-exid 38919  df-mgmOLD 38923  df-sgrOLD 38935  df-mndo 38941  df-rngo 38969

This theorem is used by:  rngoueqz  39014  rngonegmn1l  39015  rngonegmn1r  39016  rngoneglmul  39017  rngonegrmul  39018  isdrngo2  39032  rngohomco  39048  rngoisocnv  39055  idlnegcl  39096  1idl  39100  0rngo  39101  smprngopr  39126  prnc  39141  isfldidl  39142  isdmn3  39148