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Theorem ispridl2 34142
Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 34174 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 34120 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟𝑋)
4 ssralv 3799 . . . . . . . . . . . . 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 755 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
71, 2idlss 34120 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠𝑋)
8 ssralv 3799 . . . . . . . . . . . . . 14 (𝑠𝑋 → (∀𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
98ralimdv 3093 . . . . . . . . . . . . 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 754 . . . . . . . . . . 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 753 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟 ∈ (Idl‘𝑅) ∧ 𝑠 ∈ (Idl‘𝑅))) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
14 r19.26-2 3195 . . . . . . . . . . 11 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
15 pm3.35 612 . . . . . . . . . . . . 13 (((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑎𝑃𝑏𝑃))
16152ralimi 3083 . . . . . . . . . . . 12 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → ∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃))
17 2ralor 3239 . . . . . . . . . . . . 13 (∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃) ↔ (∀𝑎𝑟 𝑎𝑃 ∨ ∀𝑏𝑠 𝑏𝑃))
18 dfss3 3725 . . . . . . . . . . . . . 14 (𝑟𝑃 ↔ ∀𝑎𝑟 𝑎𝑃)
19 dfss3 3725 . . . . . . . . . . . . . 14 (𝑠𝑃 ↔ ∀𝑏𝑠 𝑏𝑃)
2018, 19orbi12i 544 . . . . . . . . . . . . 13 ((𝑟𝑃𝑠𝑃) ↔ (∀𝑎𝑟 𝑎𝑃 ∨ ∀𝑏𝑠 𝑏𝑃))
2117, 20sylbb2 228 . . . . . . . . . . . 12 (∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃) → (𝑟𝑃𝑠𝑃))
2216, 21syl 17 . . . . . . . . . . 11 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑟𝑃𝑠𝑃))
2314, 22sylbir 225 . . . . . . . . . 10 ((∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 ∧ ∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → (𝑟𝑃𝑠𝑃))
2423expcom 450 . . . . . . . . 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 3104 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))))
2726ex 449 . . . . . 6 (𝑅 ∈ RingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
2827adantrd 485 . . . . 5 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
2928imdistand 730 . . . 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 1074 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
31 df-3an 1074 . . . 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 285 . . 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 34138 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑎𝑟𝑏𝑠 (𝑎𝐻𝑏) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
3532, 34sylibrd 249 . 2 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
3635imp 444 1 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wss 3707  ran crn 5259  cfv 6041  (class class class)co 6805  1st c1st 7323  2nd c2nd 7324  RingOpscrngo 33998  Idlcidl 34111  PrIdlcpridl 34112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-idl 34114  df-pridl 34115
This theorem is referenced by:  ispridlc  34174
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