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Theorem pridlc 38057
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
pridlc (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))

Proof of Theorem pridlc
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridlc.1 . . . . 5 𝐺 = (1st𝑅)
2 ispridlc.2 . . . . 5 𝐻 = (2nd𝑅)
3 ispridlc.3 . . . . 5 𝑋 = ran 𝐺
41, 2, 3ispridlc 38056 . . . 4 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
54biimpa 476 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
65simp3d 1143 . 2 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
7 oveq1 7437 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
87eleq1d 2823 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9 eleq1 2826 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 916 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 344 . . . . . 6 (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 oveq2 7438 . . . . . . . 8 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1312eleq1d 2823 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14 eleq1 2826 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 915 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 344 . . . . . 6 (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3632 . . . . 5 ((𝐴𝑋𝐵𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1817com12 32 . . . 4 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1918expd 415 . . 3 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1348 . 2 ((∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
216, 20sylan 580 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  ran crn 5689  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  CRingOpsccring 37979  Idlcidl 37993  PrIdlcpridl 37994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-grpo 30521  df-gid 30522  df-ginv 30523  df-ablo 30573  df-ass 37829  df-exid 37831  df-mgmOLD 37835  df-sgrOLD 37847  df-mndo 37853  df-rngo 37881  df-com2 37976  df-crngo 37980  df-idl 37996  df-pridl 37997  df-igen 38046
This theorem is referenced by:  pridlc2  38058
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