Theorem idlsubcl 33822

Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
idlsubcl.1	⊢ 𝐺 = (1st ‘𝑅)
idlsubcl.2	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
idlsubcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Proof of Theorem idlsubcl

Step	Hyp	Ref	Expression
1		idlsubcl.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2622	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlcl 33816	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
4	1, 2	idlcl 33816	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵 ∈ 𝐼) → 𝐵 ∈ ran 𝐺)
5	3, 4	anim12da 33506	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺))
6		eqid 2622	. . . . . 6 ⊢ (inv‘𝐺) = (inv‘𝐺)
7		idlsubcl.2	. . . . . 6 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
8	1, 2, 6, 7	rngosub 33729	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
9	8	3expb 1266	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
10	9	adantlr 751	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
11	5, 10	syldan 487	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
12		simprl 794	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → 𝐴 ∈ 𝐼)
13	1, 6	idlnegcl 33821	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵 ∈ 𝐼) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
14	13	adantrl 752	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
15	12, 14	jca 554	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴 ∈ 𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼))
16	1	idladdcl 33818	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
17	15, 16	syldan 487	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
18	11, 17	eqeltrd 2701	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ran crn 5115  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  invcgn 27345   /_𝑔 cgs 27346  RingOpscrngo 33693  Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-idl 33809

This theorem is referenced by: (None)