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Theorem ispridl2 38024
Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38056 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 38002 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑟 ∈ (Idl‘𝑅)) → 𝑟𝑋)
4 ssralv 4063 . . . . . . . . . . . . 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 38002 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idl‘𝑅)) → 𝑠𝑋)
8 ssralv 4063 . . . . . . . . . . . . . 14 (𝑠𝑋 → (∀𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → ∀𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
98ralimdv 3166 . . . . . . . . . . . . 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 3135 . . . . . . . . . . 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 3120 . . . . . . . . . . . 12 (∀𝑎𝑟𝑏𝑠 ((𝑎𝐻𝑏) ∈ 𝑃 ∧ ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → ∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃))
17 2ralor 3228 . . . . . . . . . . . . 13 (∀𝑎𝑟𝑏𝑠 (𝑎𝑃𝑏𝑃) ↔ (∀𝑎𝑟 𝑎𝑃 ∨ ∀𝑏𝑠 𝑏𝑃))
18 dfss3 3983 . . . . . . . . . . . . . 14 (𝑟𝑃 ↔ ∀𝑎𝑟 𝑎𝑃)
19 dfss3 3983 . . . . . . . . . . . . . 14 (𝑠𝑃 ↔ ∀𝑏𝑠 𝑏𝑃)
2018, 19orbi12i 914 . . . . . . . . . . . . 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 3208 . . . . . . 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 1088 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
31 df-3an 1088 . . . 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 38020 . . 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 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wss 3962  ran crn 5689  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  RingOpscrngo 37880  Idlcidl 37993  PrIdlcpridl 37994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-idl 37996  df-pridl 37997
This theorem is referenced by:  ispridlc  38056
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