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Theorem idlsubcl 36485
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 2737 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 36479 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ ran 𝐺)
41, 2idlcl 36479 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ 𝐡 ∈ ran 𝐺)
53, 4anim12dan 620 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺))
6 eqid 2737 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔 β€˜πΊ)
81, 2, 6, 7rngosub 36392 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
983expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
109adantlr 714 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐡 ∈ ran 𝐺)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
115, 10syldan 592 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
12 simprl 770 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ 𝐴 ∈ 𝐼)
131, 6idlnegcl 36484 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐡 ∈ 𝐼) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1413adantrl 715 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)
1512, 14jca 513 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼))
161idladdcl 36481 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1715, 16syldan 592 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 𝐼)) β†’ (𝐴𝐺((invβ€˜πΊ)β€˜π΅)) ∈ 𝐼)
1811, 17eqeltrd 2838 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 397   = wceq 1542   ∈ wcel 2107  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  invcgn 29436   /𝑔 cgs 29437  RingOpscrngo 36356  Idlcidl 36469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-grpo 29438  df-gid 29439  df-ginv 29440  df-gdiv 29441  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-idl 36472
This theorem is referenced by: (None)
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