Theorem 1idl 38338

Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

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 38328	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	adantr 480	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋)
5		1idl.2	. . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅)
6	1	rneqi 5884	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7	2, 6	eqtri 2760	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
8		1idl.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
9	5, 7, 8	rngolidm 38249	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥)
10	9	ad2ant2rl 750	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥)
11	1, 5, 2	idlrmulcl 38333	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
12	10, 11	eqeltrrd 2838	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼)
13	12	expr 456	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼))
14	13	ssrdv 3928	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼)
15	4, 14	eqssd 3940	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋)
16	15	ex 412	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋))
17	7, 5, 8	rngo1cl 38251	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18	17	adantr 480	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋)
19		eleq2 2826	. . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋))
20	18, 19	syl5ibrcom 247	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼))
21	16, 20	impbid 212	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ran crn 5623  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  GIdcgi 30550  RingOpscrngo 38206  Idlcidl 38319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-riota 7315  df-ov 7361  df-1st 7933  df-2nd 7934  df-grpo 30553  df-gid 30554  df-ablo 30605  df-ass 38155  df-exid 38157  df-mgmOLD 38161  df-sgrOLD 38173  df-mndo 38179  df-rngo 38207  df-idl 38322

This theorem is referenced by:  0rngo  38339  divrngidl  38340  maxidln1  38356