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Theorem idlsubcl 37537
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 2728 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 37531 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ ran 𝐺)
41, 2idlcl 37531 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ 𝐡 ∈ ran 𝐺)
53, 4anim12dan 617 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺))
6 eqid 2728 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔 β€˜πΊ)
81, 2, 6, 7rngosub 37444 . . . . 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 713 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
115, 10syldan 589 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
12 simprl 769 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ 𝐴 ∈ 𝐼)
131, 6idlnegcl 37536 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1413adantrl 714 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1512, 14jca 510 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼))
161idladdcl 37533 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1715, 16syldan 589 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1811, 17eqeltrd 2829 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 394   = wceq 1533   ∈ wcel 2098  ran crn 5683  β€˜cfv 6553  (class class class)co 7426  1st c1st 7999  invcgn 30329   /𝑔 cgs 30330  RingOpscrngo 37408  Idlcidl 37521
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-grpo 30331  df-gid 30332  df-ginv 30333  df-gdiv 30334  df-ablo 30383  df-ass 37357  df-exid 37359  df-mgmOLD 37363  df-sgrOLD 37375  df-mndo 37381  df-rngo 37409  df-idl 37524
This theorem is referenced by: (None)
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