Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispridlc Structured version   Visualization version   GIF version

Theorem ispridlc 38077
Description: The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ispridlc.1 𝐺 = (1st𝑅)
ispridlc.2 𝐻 = (2nd𝑅)
ispridlc.3 𝑋 = ran 𝐺
Assertion
Ref Expression
ispridlc (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Distinct variable groups:   𝑅,𝑎,𝑏   𝑃,𝑎,𝑏   𝑋,𝑎,𝑏   𝐻,𝑎,𝑏
Allowed substitution hints:   𝐺(𝑎,𝑏)

Proof of Theorem ispridlc
Dummy variables 𝑥 𝑦 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngorngo 38007 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 ispridlc.1 . . . . 5 𝐺 = (1st𝑅)
3 ispridlc.2 . . . . 5 𝐻 = (2nd𝑅)
4 ispridlc.3 . . . . 5 𝑋 = ran 𝐺
52, 3, 4ispridl 38041 . . . 4 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
61, 5syl 17 . . 3 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)))))
7 snssi 4808 . . . . . . . . . . . . 13 (𝑎𝑋 → {𝑎} ⊆ 𝑋)
82, 4igenidl 38070 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
91, 7, 8syl2an 596 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
109adantrr 717 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
11 snssi 4808 . . . . . . . . . . . . 13 (𝑏𝑋 → {𝑏} ⊆ 𝑋)
122, 4igenidl 38070 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
131, 11, 12syl2an 596 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
1413adantrl 716 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
15 raleq 3323 . . . . . . . . . . . . 13 (𝑟 = (𝑅 IdlGen {𝑎}) → (∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃))
16 sseq1 4009 . . . . . . . . . . . . . 14 (𝑟 = (𝑅 IdlGen {𝑎}) → (𝑟𝑃 ↔ (𝑅 IdlGen {𝑎}) ⊆ 𝑃))
1716orbi1d 917 . . . . . . . . . . . . 13 (𝑟 = (𝑅 IdlGen {𝑎}) → ((𝑟𝑃𝑠𝑃) ↔ ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃)))
1815, 17imbi12d 344 . . . . . . . . . . . 12 (𝑟 = (𝑅 IdlGen {𝑎}) → ((∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) ↔ (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃))))
19 raleq 3323 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
2019ralbidv 3178 . . . . . . . . . . . . 13 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
21 sseq1 4009 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (𝑠𝑃 ↔ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))
2221orbi2d 916 . . . . . . . . . . . . 13 (𝑠 = (𝑅 IdlGen {𝑏}) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃) ↔ ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃)))
2320, 22imbi12d 344 . . . . . . . . . . . 12 (𝑠 = (𝑅 IdlGen {𝑏}) → ((∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃)) ↔ (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))))
2418, 23rspc2v 3633 . . . . . . . . . . 11 (((𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅) ∧ (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))))
2510, 14, 24syl2anc 584 . . . . . . . . . 10 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))))
2625adantlr 715 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))))
272, 3, 4prnc 38074 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)})
28 df-rab 3437 . . . . . . . . . . . . . . . . . . 19 {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)} = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))}
2927, 28eqtrdi 2793 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))})
3029eqabrd 2884 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
3130adantrr 717 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
322, 3, 4prnc 38074 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)})
33 df-rab 3437 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)} = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))}
3432, 33eqtrdi 2793 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))})
3534eqabrd 2884 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3635adantrl 716 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3731, 36anbi12d 632 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3837adantlr 715 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3938adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
40 reeanv 3229 . . . . . . . . . . . . . . . 16 (∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) ↔ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))
4140anbi2i 623 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋𝑦𝑋) ∧ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
42 an4 656 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
4341, 42bitri 275 . . . . . . . . . . . . . 14 (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
442, 3, 4crngm4 38010 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
45443com23 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
46453expa 1119 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4746adantllr 719 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4847adantlr 715 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
492, 3, 4rngocl 37908 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
501, 49syl3an1 1164 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ CRingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
51503expb 1121 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5251adantlr 715 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5352adantlr 715 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
542, 3, 4idllmulcl 38027 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
551, 54sylanl1 680 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5655anassrs 467 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝐻𝑠) ∈ 𝑋) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5753, 56syldan 591 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5857adantllr 719 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5948, 58eqeltrrd 2842 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃)
60 oveq12 7440 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
6160eleq1d 2826 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → ((𝑥𝐻𝑦) ∈ 𝑃 ↔ ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃))
6259, 61syl5ibrcom 247 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6362rexlimdvva 3213 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6463adantld 490 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6543, 64biimtrrid 243 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6639, 65sylbid 240 . . . . . . . . . . . 12 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) → (𝑥𝐻𝑦) ∈ 𝑃))
6766ralrimivv 3200 . . . . . . . . . . 11 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃)
6867ex 412 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐻𝑏) ∈ 𝑃 → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
692, 4igenss 38069 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
701, 7, 69syl2an 596 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
71 vex 3484 . . . . . . . . . . . . . . . 16 𝑎 ∈ V
7271snss 4785 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝑅 IdlGen {𝑎}) ↔ {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
7370, 72sylibr 234 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
7473adantrr 717 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
75 ssel 3977 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 → (𝑎 ∈ (𝑅 IdlGen {𝑎}) → 𝑎𝑃))
7674, 75syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑎𝑃))
772, 4igenss 38069 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
781, 11, 77syl2an 596 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
79 vex 3484 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
8079snss 4785 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝑅 IdlGen {𝑏}) ↔ {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
8178, 80sylibr 234 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
8281adantrl 716 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
83 ssel 3977 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑏}) ⊆ 𝑃 → (𝑏 ∈ (𝑅 IdlGen {𝑏}) → 𝑏𝑃))
8482, 83syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑏}) ⊆ 𝑃𝑏𝑃))
8576, 84orim12d 967 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃) → (𝑎𝑃𝑏𝑃)))
8685adantlr 715 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃) → (𝑎𝑃𝑏𝑃)))
8768, 86imim12d 81 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃)) → ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
8826, 87syld 47 . . . . . . . 8 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
8988ralrimdvva 3211 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
9089ex 412 . . . . . 6 (𝑅 ∈ CRingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9190adantrd 491 . . . . 5 (𝑅 ∈ CRingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9291imdistand 570 . . . 4 (𝑅 ∈ CRingOps → (((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))) → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
93 df-3an 1089 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))))
94 df-3an 1089 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
9592, 93, 943imtr4g 296 . . 3 (𝑅 ∈ CRingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃))) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
966, 95sylbid 240 . 2 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
972, 3, 4ispridl2 38045 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
9897ex 412 . . 3 (𝑅 ∈ RingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
991, 98syl 17 . 2 (𝑅 ∈ CRingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) → 𝑃 ∈ (PrIdl‘𝑅)))
10096, 99impbid 212 1 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  {csn 4626  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  RingOpscrngo 37901  CRingOpsccring 38000  Idlcidl 38014  PrIdlcpridl 38015   IdlGen cigen 38066
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-rep 5279  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-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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-int 4947  df-iun 4993  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-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-com2 37997  df-crngo 38001  df-idl 38017  df-pridl 38018  df-igen 38067
This theorem is referenced by:  pridlc  38078  isdmn3  38081
  Copyright terms: Public domain W3C validator