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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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