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Theorem pridlc3 34793
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 34720 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 eldifi 3987 . . . . 5 (𝐴 ∈ (𝑋𝑃) → 𝐴𝑋)
3 eldifi 3987 . . . . 5 (𝐵 ∈ (𝑋𝑃) → 𝐵𝑋)
42, 3anim12i 603 . . . 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 34621 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
983expb 1100 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
101, 4, 9syl2an 586 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
1110adantlr 702 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12 eldifn 3988 . . . 4 (𝐵 ∈ (𝑋𝑃) → ¬ 𝐵𝑃)
1312ad2antll 716 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → ¬ 𝐵𝑃)
145, 6, 7pridlc2 34792 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵𝑃)
15143exp2 1334 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋𝑃) → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))))
1615imp32 411 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃𝐵𝑃))
1716con3d 150 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋)) → (¬ 𝐵𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
183, 17sylanr2 670 . . 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 3834 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 387   = wceq 1507  wcel 2050  cdif 3820  ran crn 5402  cfv 6182  (class class class)co 6970  1st c1st 7493  2nd c2nd 7494  RingOpscrngo 34614  CRingOpsccring 34713  PrIdlcpridl 34728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7495  df-2nd 7496  df-grpo 28041  df-gid 28042  df-ginv 28043  df-ablo 28093  df-ass 34563  df-exid 34565  df-mgmOLD 34569  df-sgrOLD 34581  df-mndo 34587  df-rngo 34615  df-com2 34710  df-crngo 34714  df-idl 34730  df-pridl 34731  df-igen 34780
This theorem is referenced by: (None)
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