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Theorem ispridl2 38045
Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38077 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ispridl2.1 𝐺 = (1st𝑅)
ispridl2.2 𝐻 = (2nd𝑅)
ispridl2.3 𝑋 = ran 𝐺
Assertion
Ref Expression
ispridl2 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
Distinct variable groups:   𝑅,𝑎,𝑏   𝑃,𝑎,𝑏   𝑋,𝑎,𝑏
Allowed substitution hints:   𝐺(𝑎,𝑏)   𝐻(𝑎,𝑏)

Proof of Theorem ispridl2
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridl2.1 . . . . . . . . . . . . . 14 𝐺 = (1st𝑅)
2 ispridl2.3 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
31, 2idlss 38023 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟𝑋)
4 ssralv 4052 . . . . . . . . . . . . 13 (𝑟𝑋 → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
53, 4syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
65adantrr 717 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
71, 2idlss 38023 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠𝑋)
8 ssralv 4052 . . . . . . . . . . . . . 14 (𝑠𝑋 → (∀𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
98ralimdv 3169 . . . . . . . . . . . . 13 (𝑠𝑋 → (∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
107, 9syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → (∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
1110adantrl 716 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
126, 11syld 47 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
1312adantlr 715 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
14 r19.26-2 3138 . . . . . . . . . . 11 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
15 pm3.35 803 . . . . . . . . . . . . 13 (((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑎𝑃𝑏𝑃))
16152ralimi 3123 . . . . . . . . . . . 12 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → ∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃))
17 2ralor 3231 . . . . . . . . . . . . 13 (∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃) ↔ (∀𝑎𝑟 𝑎𝑃 ∨ ∀𝑏𝑠 𝑏𝑃))
18 dfss3 3972 . . . . . . . . . . . . . 14 (𝑟𝑃 ↔ ∀𝑎𝑟 𝑎𝑃)
19 dfss3 3972 . . . . . . . . . . . . . 14 (𝑠𝑃 ↔ ∀𝑏𝑠 𝑏𝑃)
2018, 19orbi12i 915 . . . . . . . . . . . . 13 ((𝑟𝑃𝑠𝑃) ↔ (∀𝑎𝑟 𝑎𝑃 ∨ ∀𝑏𝑠 𝑏𝑃))
2117, 20sylbb2 238 . . . . . . . . . . . 12 (∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃) → (𝑟𝑃𝑠𝑃))
2216, 21syl 17 . . . . . . . . . . 11 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑟𝑃𝑠𝑃))
2314, 22sylbir 235 . . . . . . . . . 10 ((∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑟𝑃𝑠𝑃))
2423expcom 413 . . . . . . . . 9 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))
2513, 24syl6 35 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))))
2625ralrimdvva 3211 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))))
2726ex 412 . . . . . 6 (𝑅 ∈ RingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
2827adantrd 491 . . . . 5 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
2928imdistand 570 . . . 4 (𝑅 ∈ RingOps → (((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
30 df-3an 1089 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
31 df-3an 1089 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))))
3229, 30, 313imtr4g 296 . . 3 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
33 ispridl2.2 . . . 4 𝐻 = (2nd𝑅)
341, 33, 2ispridl 38041 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
3532, 34sylibrd 259 . 2 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
3635imp 406 1 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  RingOpscrngo 37901  Idlcidl 38014  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-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-rab 3437  df-v 3482  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-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-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-idl 38017  df-pridl 38018
This theorem is referenced by:  ispridlc  38077
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