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Theorem pridlc3 36936
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
ispridlc.1 𝐺 = (1st β€˜π‘…)
ispridlc.2 𝐻 = (2nd β€˜π‘…)
ispridlc.3 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlc3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ (𝑋 βˆ– 𝑃))
Proof of Theorem pridlc3
StepHypRef Expression
1 crngorngo 36863 . . . 4 (𝑅 ∈ CRingOps β†’ 𝑅 ∈ RingOps)
2 eldifi 4126 . . . . 5 (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐴 ∈ 𝑋)
3 eldifi 4126 . . . . 5 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐡 ∈ 𝑋)
42, 3anim12i 613 . . . 4 ((𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
5 ispridlc.1 . . . . . 6 𝐺 = (1st β€˜π‘…)
6 ispridlc.2 . . . . . 6 𝐻 = (2nd β€˜π‘…)
7 ispridlc.3 . . . . . 6 𝑋 = ran 𝐺
85, 6, 7rngocl 36764 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
983expb 1120 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
101, 4, 9syl2an 596 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
1110adantlr 713 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
12 eldifn 4127 . . . 4 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ Β¬ 𝐡 ∈ 𝑃)
1312ad2antll 727 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ 𝐡 ∈ 𝑃)
145, 6, 7pridlc2 36935 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ 𝐡 ∈ 𝑃)
15143exp2 1354 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))))
1615imp32 419 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))
1716con3d 152 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋)) β†’ (Β¬ 𝐡 ∈ 𝑃 β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃))
183, 17sylanr2 681 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (Β¬ 𝐡 ∈ 𝑃 β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃))
1913, 18mpd 15 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃)
2011, 19eldifd 3959 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ (𝑋 βˆ– 𝑃))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3945  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  RingOpscrngo 36757  CRingOpsccring 36856  PrIdlcpridl 36871
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ginv 29743  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758  df-com2 36853  df-crngo 36857  df-idl 36873  df-pridl 36874  df-igen 36923
This theorem is referenced by: (None)
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