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Theorem ispridlc 36529
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 36459 . . . 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 36493 . . . 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 4768 . . . . . . . . . . . . 13 (𝑎𝑋 → {𝑎} ⊆ 𝑋)
82, 4igenidl 36522 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
91, 7, 8syl2an 596 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
109adantrr 715 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑎}) ∈ (Idl‘𝑅))
11 snssi 4768 . . . . . . . . . . . . 13 (𝑏𝑋 → {𝑏} ⊆ 𝑋)
122, 4igenidl 36522 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
131, 11, 12syl2an 596 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
1413adantrl 714 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑅 IdlGen {𝑏}) ∈ (Idl‘𝑅))
15 raleq 3309 . . . . . . . . . . . . 13 (𝑟 = (𝑅 IdlGen {𝑎}) → (∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃))
16 sseq1 3969 . . . . . . . . . . . . . 14 (𝑟 = (𝑅 IdlGen {𝑎}) → (𝑟𝑃 ↔ (𝑅 IdlGen {𝑎}) ⊆ 𝑃))
1716orbi1d 915 . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
2019ralbidv 3174 . . . . . . . . . . . . 13 (𝑠 = (𝑅 IdlGen {𝑏}) → (∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
21 sseq1 3969 . . . . . . . . . . . . . 14 (𝑠 = (𝑅 IdlGen {𝑏}) → (𝑠𝑃 ↔ (𝑅 IdlGen {𝑏}) ⊆ 𝑃))
2221orbi2d 914 . . . . . . . . . . . . 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 3590 . . . . . . . . . . 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 713 . . . . . . . . 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 36526 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)})
28 df-rab 3408 . . . . . . . . . . . . . . . . . . 19 {𝑥𝑋 ∣ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)} = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))}
2927, 28eqtrdi 2792 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑅 IdlGen {𝑎}) = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))})
3029eqabd 2880 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
3130adantrr 715 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑥 ∈ (𝑅 IdlGen {𝑎}) ↔ (𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎))))
322, 3, 4prnc 36526 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)})
33 df-rab 3408 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑋 ∣ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)} = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))}
3432, 33eqtrdi 2792 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑅 IdlGen {𝑏}) = {𝑦 ∣ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))})
3534eqabd 2880 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3635adantrl 714 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (𝑦 ∈ (𝑅 IdlGen {𝑏}) ↔ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
3731, 36anbi12d 631 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3837adantlr 713 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏)))))
3938adantr 481 . . . . . . . . . . . . 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 3217 . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ (∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎) ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
4341, 42bitri 274 . . . . . . . . . . . . . 14 (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) ↔ ((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))))
442, 3, 4crngm4 36462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
45443com23 1126 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
46453expa 1118 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4746adantllr 717 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
4847adantlr 713 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
492, 3, 4rngocl 36360 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
501, 49syl3an1 1163 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ CRingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐻𝑠) ∈ 𝑋)
51503expb 1120 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ CRingOps ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5251adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
5352adantlr 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → (𝑟𝐻𝑠) ∈ 𝑋)
542, 3, 4idllmulcl 36479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
551, 54sylanl1 678 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ ((𝑎𝐻𝑏) ∈ 𝑃 ∧ (𝑟𝐻𝑠) ∈ 𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5655anassrs 468 . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑠)𝐻(𝑎𝐻𝑏)) ∈ 𝑃)
5948, 58eqeltrrd 2839 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃)
60 oveq12 7366 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) = ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)))
6160eleq1d 2822 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → ((𝑥𝐻𝑦) ∈ 𝑃 ↔ ((𝑟𝐻𝑎)𝐻(𝑠𝐻𝑏)) ∈ 𝑃))
6259, 61syl5ibrcom 246 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6362rexlimdvva 3205 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏)) → (𝑥𝐻𝑦) ∈ 𝑃))
6463adantld 491 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋𝑦𝑋) ∧ ∃𝑟𝑋𝑠𝑋 (𝑥 = (𝑟𝐻𝑎) ∧ 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6543, 64biimtrrid 242 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → (((𝑥𝑋 ∧ ∃𝑟𝑋 𝑥 = (𝑟𝐻𝑎)) ∧ (𝑦𝑋 ∧ ∃𝑠𝑋 𝑦 = (𝑠𝐻𝑏))) → (𝑥𝐻𝑦) ∈ 𝑃))
6639, 65sylbid 239 . . . . . . . . . . . 12 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ((𝑥 ∈ (𝑅 IdlGen {𝑎}) ∧ 𝑦 ∈ (𝑅 IdlGen {𝑏})) → (𝑥𝐻𝑦) ∈ 𝑃))
6766ralrimivv 3195 . . . . . . . . . . 11 ((((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑎𝐻𝑏) ∈ 𝑃) → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃)
6867ex 413 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐻𝑏) ∈ 𝑃 → ∀𝑥 ∈ (𝑅 IdlGen {𝑎})∀𝑦 ∈ (𝑅 IdlGen {𝑏})(𝑥𝐻𝑦) ∈ 𝑃))
692, 4igenss 36521 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑎} ⊆ 𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
701, 7, 69syl2an 596 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
71 vex 3449 . . . . . . . . . . . . . . . 16 𝑎 ∈ V
7271snss 4746 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝑅 IdlGen {𝑎}) ↔ {𝑎} ⊆ (𝑅 IdlGen {𝑎}))
7370, 72sylibr 233 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑎𝑋) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
7473adantrr 715 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (𝑅 IdlGen {𝑎}))
75 ssel 3937 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑎}) ⊆ 𝑃 → (𝑎 ∈ (𝑅 IdlGen {𝑎}) → 𝑎𝑃))
7674, 75syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑎}) ⊆ 𝑃𝑎𝑃))
772, 4igenss 36521 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ {𝑏} ⊆ 𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
781, 11, 77syl2an 596 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
79 vex 3449 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
8079snss 4746 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝑅 IdlGen {𝑏}) ↔ {𝑏} ⊆ (𝑅 IdlGen {𝑏}))
8178, 80sylibr 233 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRingOps ∧ 𝑏𝑋) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
8281adantrl 714 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (𝑅 IdlGen {𝑏}))
83 ssel 3937 . . . . . . . . . . . . 13 ((𝑅 IdlGen {𝑏}) ⊆ 𝑃 → (𝑏 ∈ (𝑅 IdlGen {𝑏}) → 𝑏𝑃))
8482, 83syl5com 31 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → ((𝑅 IdlGen {𝑏}) ⊆ 𝑃𝑏𝑃))
8576, 84orim12d 963 . . . . . . . . . . 11 ((𝑅 ∈ CRingOps ∧ (𝑎𝑋𝑏𝑋)) → (((𝑅 IdlGen {𝑎}) ⊆ 𝑃 ∨ (𝑅 IdlGen {𝑏}) ⊆ 𝑃) → (𝑎𝑃𝑏𝑃)))
8685adantlr 713 . . . . . . . . . 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 3203 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (Idl‘𝑅)) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
9089ex 413 . . . . . 6 (𝑅 ∈ CRingOps → (𝑃 ∈ (Idl‘𝑅) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9190adantrd 492 . . . . 5 (𝑅 ∈ CRingOps → ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋) → (∀𝑟 ∈ (Idl‘𝑅)∀𝑠 ∈ (Idl‘𝑅)(∀𝑥𝑟𝑦𝑠 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑟𝑃𝑠𝑃)) → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
9291imdistand 571 . . . 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 295 . . 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 239 . 2 (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
972, 3, 4ispridl2 36497 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
9897ex 413 . . 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 211 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  wss 3910  {csn 4586  ran crn 5634  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  RingOpscrngo 36353  CRingOpsccring 36452  Idlcidl 36466  PrIdlcpridl 36467   IdlGen cigen 36518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-grpo 29435  df-gid 29436  df-ginv 29437  df-ablo 29487  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326  df-rngo 36354  df-com2 36449  df-crngo 36453  df-idl 36469  df-pridl 36470  df-igen 36519
This theorem is referenced by:  pridlc  36530  isdmn3  36533
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