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Theorem ispridlc 34101
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 34031 . . . 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 34065 . . . 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 4447 . . . . . . . . . . . . 13 (𝑎𝑋 → {𝑎} ⊆ 𝑋)
82, 4igenidl 34094 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
91, 7, 8syl2an 495 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
109adantrr 755 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
11 snssi 4447 . . . . . . . . . . . . 13 (𝑏𝑋 → {𝑏} ⊆ 𝑋)
122, 4igenidl 34094 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
131, 11, 12syl2an 495 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
1413adantrl 754 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
15 raleq 3241 . . . . . . . . . . . . 13 (𝑟 = (𝑅 IdlGen {𝑎}) → (∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃))
16 sseq1 3732 . . . . . . . . . . . . . 14 (𝑟 = (𝑅 IdlGen {𝑎}) → (𝑟𝑃 ↔ (𝑅 IdlGen {𝑎}) ⊆ 𝑃))
1716orbi1d 741 . . . . . . . . . . . . 13 (𝑟 = (𝑅 IdlGen {𝑎}) → ((𝑟𝑃𝑠𝑃) ↔ ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃)))
1815, 17imbi12d 333 . . . . . . . . . . . 12 (𝑟 = (𝑅 IdlGen {𝑎}) → ((∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) ↔ (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃))))
19 raleq 3241 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
2019ralbidv 3088 . . . . . . . . . . . . 13 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
21 sseq1 3732 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (𝑠𝑃 ↔ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))
2221orbi2d 740 . . . . . . . . . . . . 13 (𝑠 = (𝑅 IdlGen {𝑏}) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑠𝑃) ↔ ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃)))
2320, 22imbi12d 333 . . . . . . . . . . . 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 3426 . . . . . . . . . . 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 696 . . . . . . . . . 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 753 . . . . . . . . 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 34098 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)})
28 df-rab 3023 . . . . . . . . . . . . . . . . . . 19 {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)} = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))}
2927, 28syl6eq 2774 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))})
3029abeq2d 2836 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
3130adantrr 755 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
322, 3, 4prnc 34098 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)})
33 df-rab 3023 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)} = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))}
3432, 33syl6eq 2774 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))})
3534abeq2d 2836 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3635adantrl 754 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3731, 36anbi12d 749 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3837adantlr 753 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3938adantr 472 . . . . . . . . . . . . 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 3209 . . . . . . . . . . . . . . . 16 (∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) ↔ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))
4140anbi2i 732 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋𝑦𝑋) ∧ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
42 an4 900 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
4341, 42bitri 264 . . . . . . . . . . . . . 14 (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
442, 3, 4crngm4 34034 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
45443com23 1120 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
46453expa 1111 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4746adantllr 757 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4847adantlr 753 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
492, 3, 4rngocl 33932 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
501, 49syl3an1 1472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ CRingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
51503expb 1113 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5251adantlr 753 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5352adantlr 753 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
542, 3, 4idllmulcl 34051 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
551, 54sylanl1 685 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5655anassrs 683 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝐻𝑠) ∈ 𝑋) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5753, 56syldan 488 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5857adantllr 757 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5948, 58eqeltrrd 2804 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃)
60 oveq12 6774 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
6160eleq1d 2788 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → ((𝑥𝐻𝑦) ∈ 𝑃 ↔ ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃))
6259, 61syl5ibrcom 237 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6362rexlimdvva 3140 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6463adantld 484 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6543, 64syl5bir 233 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6639, 65sylbid 230 . . . . . . . . . . . 12 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) → (𝑥𝐻𝑦) ∈ 𝑃))
6766ralrimivv 3072 . . . . . . . . . . 11 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃)
6867ex 449 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐻𝑏) ∈ 𝑃 → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
692, 4igenss 34093 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
701, 7, 69syl2an 495 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
71 vex 3307 . . . . . . . . . . . . . . . 16 𝑎 ∈ V
7271snss 4423 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝑅 IdlGen {𝑎}) ↔ {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
7370, 72sylibr 224 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
7473adantrr 755 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
75 ssel 3703 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 → (𝑎 ∈ (𝑅 IdlGen {𝑎}) → 𝑎𝑃))
7674, 75syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑎𝑃))
772, 4igenss 34093 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
781, 11, 77syl2an 495 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
79 vex 3307 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
8079snss 4423 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝑅 IdlGen {𝑏}) ↔ {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
8178, 80sylibr 224 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
8281adantrl 754 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
83 ssel 3703 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑏}) ⊆ 𝑃 → (𝑏 ∈ (𝑅 IdlGen {𝑏}) → 𝑏𝑃))
8482, 83syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑏}) ⊆ 𝑃𝑏𝑃))
8576, 84orim12d 919 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃) → (𝑎𝑃𝑏𝑃)))
8685adantlr 753 . . . . . . . . . 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 3076 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
9089ex 449 . . . . . 6 (𝑅 ∈ CRingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9190adantrd 485 . . . . 5 (𝑅 ∈ CRingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9291imdistand 730 . . . 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 1074 . . . 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 1074 . . . 4 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
9592, 93, 943imtr4g 285 . . 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 230 . 2 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
972, 3, 4ispridl2 34069 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
9897ex 449 . . 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 202 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 196  wo 382  wa 383  w3a 1072   = wceq 1596  wcel 2103  {cab 2710  wne 2896  wral 3014  wrex 3015  {crab 3018  wss 3680  {csn 4285  ran crn 5219  cfv 6001  (class class class)co 6765  1st c1st 7283  2nd c2nd 7284  RingOpscrngo 33925  CRingOpsccring 34024  Idlcidl 34038  PrIdlcpridl 34039   IdlGen cigen 34090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-grpo 27577  df-gid 27578  df-ginv 27579  df-ablo 27629  df-ass 33874  df-exid 33876  df-mgmOLD 33880  df-sgrOLD 33892  df-mndo 33898  df-rngo 33926  df-com2 34021  df-crngo 34025  df-idl 34041  df-pridl 34042  df-igen 34091
This theorem is referenced by:  pridlc  34102  isdmn3  34105
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