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Theorem pridlc3 38080
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 38007 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 eldifi 4131 . . . . 5 (𝐴 ∈ (𝑋𝑃) → 𝐴𝑋)
3 eldifi 4131 . . . . 5 (𝐵 ∈ (𝑋𝑃) → 𝐵𝑋)
42, 3anim12i 613 . . . 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 37908 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
983expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
101, 4, 9syl2an 596 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
1110adantlr 715 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12 eldifn 4132 . . . 4 (𝐵 ∈ (𝑋𝑃) → ¬ 𝐵𝑃)
1312ad2antll 729 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → ¬ 𝐵𝑃)
145, 6, 7pridlc2 38079 . . . . . . 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 418 . . . . 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 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 3962 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 395   = wceq 1540  wcel 2108  cdif 3948  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  RingOpscrngo 37901  CRingOpsccring 38000  PrIdlcpridl 38015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-com2 37997  df-crngo 38001  df-idl 38017  df-pridl 38018  df-igen 38067
This theorem is referenced by: (None)
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