Theorem 1idl 39003

Description: Obsolete theorem, use 2idl1el 21549 instead. Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
1idl.1	⊢ 𝐺 = (1st ‘𝑅)
1idl.2	⊢ 𝐻 = (2nd ‘𝑅)
1idl.3	⊢ 𝑋 = ran 𝐺
1idl.4	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
1idl	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Proof of Theorem 1idl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1idl.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		1idl.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
3	1, 2	idlss 38993	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	adantr 486	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋)
5		1idl.2	. . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅)
6	1	rneqi 5918	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7	2, 6	eqtri 2783	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
8		1idl.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
9	5, 7, 8	rngolidm 38914	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥)
10	9	ad2ant2rl 762	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥)
11	1, 5, 2	idlrmulcl 38998	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
12	10, 11	eqeltrrd 2861	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼)
13	12	expr 462	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼))
14	13	ssrdv 3914	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼)
15	4, 14	eqssd 3925	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋)
16	15	ex 418	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋))
17	7, 5, 8	rngo1cl 38916	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18	17	adantr 486	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋)
19		eleq2 2849	. . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋))
20	18, 19	syl5ibrcom 250	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼))
21	16, 20	impbid 215	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  GIdcgi 31092  RingOpscrngo 38871  Idlcidl 38984

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-grpo 31095  df-gid 31096  df-ablo 31147  df-ass 38820  df-exid 38822  df-mgmOLD 38826  df-sgrOLD 38838  df-mndo 38844  df-rngo 38872  df-idl 38987

This theorem is used by:  0rngo  39004  divrngidl  39005  maxidln1  39021