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

Theorem pridlc3 33042
 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 32969 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 eldifi 3694 . . . . 5 (𝐴 ∈ (𝑋𝑃) → 𝐴𝑋)
3 eldifi 3694 . . . . 5 (𝐵 ∈ (𝑋𝑃) → 𝐵𝑋)
42, 3anim12i 588 . . . 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 32870 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
983expb 1258 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
101, 4, 9syl2an 493 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
1110adantlr 747 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12 eldifn 3695 . . . 4 (𝐵 ∈ (𝑋𝑃) → ¬ 𝐵𝑃)
1312ad2antll 761 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → ¬ 𝐵𝑃)
145, 6, 7pridlc2 33041 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵𝑃)
15143exp2 1277 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋𝑃) → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))))
1615imp32 448 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))
1716con3d 147 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → (¬ 𝐵𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
183, 17sylanr2 683 . . 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 3551 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 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  RingOpscrngo 32863  CRingOpsccring 32962  PrIdlcpridl 32977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-com2 32959  df-crngo 32963  df-idl 32979  df-pridl 32980  df-igen 33029 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator