Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pridlc3 Structured version   Visualization version   GIF version

Theorem pridlc3 36847
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
ispridlc.1 𝐺 = (1st𝑅)
ispridlc.2 𝐻 = (2nd𝑅)
ispridlc.3 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlc3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋𝑃))

Proof of Theorem pridlc3
StepHypRef Expression
1 crngorngo 36774 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 eldifi 4124 . . . . 5 (𝐴 ∈ (𝑋𝑃) → 𝐴𝑋)
3 eldifi 4124 . . . . 5 (𝐵 ∈ (𝑋𝑃) → 𝐵𝑋)
42, 3anim12i 614 . . . 4 ((𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃)) → (𝐴𝑋𝐵𝑋))
5 ispridlc.1 . . . . . 6 𝐺 = (1st𝑅)
6 ispridlc.2 . . . . . 6 𝐻 = (2nd𝑅)
7 ispridlc.3 . . . . . 6 𝑋 = ran 𝐺
85, 6, 7rngocl 36675 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
983expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
101, 4, 9syl2an 597 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
1110adantlr 714 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12 eldifn 4125 . . . 4 (𝐵 ∈ (𝑋𝑃) → ¬ 𝐵𝑃)
1312ad2antll 728 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → ¬ 𝐵𝑃)
145, 6, 7pridlc2 36846 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵𝑃)
15143exp2 1355 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋𝑃) → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))))
1615imp32 420 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))
1716con3d 152 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → (¬ 𝐵𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
183, 17sylanr2 682 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (¬ 𝐵𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
1913, 18mpd 15 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → ¬ (𝐴𝐻𝐵) ∈ 𝑃)
2011, 19eldifd 3957 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋𝑃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  cdif 3943  ran crn 5673  cfv 6535  (class class class)co 7396  1st c1st 7960  2nd c2nd 7961  RingOpscrngo 36668  CRingOpsccring 36767  PrIdlcpridl 36782
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 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-grpo 29711  df-gid 29712  df-ginv 29713  df-ablo 29763  df-ass 36617  df-exid 36619  df-mgmOLD 36623  df-sgrOLD 36635  df-mndo 36641  df-rngo 36669  df-com2 36764  df-crngo 36768  df-idl 36784  df-pridl 36785  df-igen 36834
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator