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Theorem pridlc 32838
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 32837 . . . 4 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
54biimpa 499 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
65simp3d 1067 . 2 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
7 oveq1 6529 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
87eleq1d 2666 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9 eleq1 2670 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 734 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 332 . . . . . 6 (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 oveq2 6530 . . . . . . . 8 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1312eleq1d 2666 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14 eleq1 2670 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 733 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 332 . . . . . 6 (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3287 . . . . 5 ((𝐴𝑋𝐵𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1817com12 32 . . . 4 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1918expd 450 . . 3 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1273 . 2 ((∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
216, 20sylan 486 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774  wral 2890  ran crn 5024  cfv 5785  (class class class)co 6522  1st c1st 7029  2nd c2nd 7030  CRingOpsccring 32760  Idlcidl 32774  PrIdlcpridl 32775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-grpo 26492  df-gid 26493  df-ginv 26494  df-ablo 26547  df-ass 32610  df-exid 32612  df-mgmOLD 32616  df-sgrOLD 32628  df-mndo 32634  df-rngo 32662  df-com2 32757  df-crngo 32761  df-idl 32777  df-pridl 32778  df-igen 32827
This theorem is referenced by:  pridlc2  32839
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