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Theorem pridlc3 37452
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 37379 . . . 4 (𝑅 ∈ CRingOps β†’ 𝑅 ∈ RingOps)
2 eldifi 4121 . . . . 5 (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐴 ∈ 𝑋)
3 eldifi 4121 . . . . 5 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐡 ∈ 𝑋)
42, 3anim12i 612 . . . 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 37280 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
983expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
101, 4, 9syl2an 595 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
1110adantlr 712 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
12 eldifn 4122 . . . 4 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ Β¬ 𝐡 ∈ 𝑃)
1312ad2antll 726 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ 𝐡 ∈ 𝑃)
145, 6, 7pridlc2 37451 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ 𝐡 ∈ 𝑃)
15143exp2 1351 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))))
1615imp32 418 . . . . 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 680 . . 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 3954 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 395   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3940  ran crn 5670  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  RingOpscrngo 37273  CRingOpsccring 37372  PrIdlcpridl 37387
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 7721
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-int 4944  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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-grpo 30251  df-gid 30252  df-ginv 30253  df-ablo 30303  df-ass 37222  df-exid 37224  df-mgmOLD 37228  df-sgrOLD 37240  df-mndo 37246  df-rngo 37274  df-com2 37369  df-crngo 37373  df-idl 37389  df-pridl 37390  df-igen 37439
This theorem is referenced by: (None)
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