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Theorem idlsubcl 37404
Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlsubcl.1 𝐺 = (1st β€˜π‘…)
idlsubcl.2 𝐷 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
idlsubcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐷𝐡) ∈ 𝐼)

Proof of Theorem idlsubcl
StepHypRef Expression
1 idlsubcl.1 . . . . 5 𝐺 = (1st β€˜π‘…)
2 eqid 2726 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 37398 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ ran 𝐺)
41, 2idlcl 37398 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ 𝐡 ∈ ran 𝐺)
53, 4anim12dan 618 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺))
6 eqid 2726 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔 β€˜πΊ)
81, 2, 6, 7rngosub 37311 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
983expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
109adantlr 712 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
115, 10syldan 590 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
12 simprl 768 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ 𝐴 ∈ 𝐼)
131, 6idlnegcl 37403 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1413adantrl 713 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1512, 14jca 511 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼))
161idladdcl 37400 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1715, 16syldan 590 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1811, 17eqeltrd 2827 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐷𝐡) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  invcgn 30253   /𝑔 cgs 30254  RingOpscrngo 37275  Idlcidl 37388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258  df-ablo 30307  df-ass 37224  df-exid 37226  df-mgmOLD 37230  df-sgrOLD 37242  df-mndo 37248  df-rngo 37276  df-idl 37391
This theorem is referenced by: (None)
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