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Theorem pridlc 35395
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 35394 . . . 4 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
54biimpa 480 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
65simp3d 1141 . 2 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
7 oveq1 7137 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
87eleq1d 2896 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9 eleq1 2899 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 914 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 348 . . . . . 6 (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 oveq2 7138 . . . . . . . 8 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1312eleq1d 2896 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14 eleq1 2899 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 913 . . . . . . 7 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 348 . . . . . 6 (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3610 . . . . 5 ((𝐴𝑋𝐵𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1817com12 32 . . . 4 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1918expd 419 . . 3 (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1346 . 2 ((∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
216, 20sylan 583 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  ran crn 5529  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  CRingOpsccring 35317  Idlcidl 35331  PrIdlcpridl 35332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-grpo 28255  df-gid 28256  df-ginv 28257  df-ablo 28307  df-ass 35167  df-exid 35169  df-mgmOLD 35173  df-sgrOLD 35185  df-mndo 35191  df-rngo 35219  df-com2 35314  df-crngo 35318  df-idl 35334  df-pridl 35335  df-igen 35384
This theorem is referenced by:  pridlc2  35396
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