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Theorem prnc 33996
Description: A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
prnc.1 𝐺 = (1st𝑅)
prnc.2 𝐻 = (2nd𝑅)
prnc.3 𝑋 = ran 𝐺
Assertion
Ref Expression
prnc ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝐴,𝑦

Proof of Theorem prnc
Dummy variables 𝑗 𝑢 𝑣 𝑤 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngorngo 33929 . . . . 5 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2 ssrab2 3720 . . . . . . 7 {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋
32a1i 11 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋)
4 prnc.1 . . . . . . . . 9 𝐺 = (1st𝑅)
5 prnc.3 . . . . . . . . 9 𝑋 = ran 𝐺
6 eqid 2651 . . . . . . . . 9 (GId‘𝐺) = (GId‘𝐺)
74, 5, 6rngo0cl 33848 . . . . . . . 8 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
87adantr 480 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (GId‘𝐺) ∈ 𝑋)
9 prnc.2 . . . . . . . . . 10 𝐻 = (2nd𝑅)
106, 5, 4, 9rngolz 33851 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
1110eqcomd 2657 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴))
12 oveq1 6697 . . . . . . . . . 10 (𝑦 = (GId‘𝐺) → (𝑦𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
1312eqeq2d 2661 . . . . . . . . 9 (𝑦 = (GId‘𝐺) → ((GId‘𝐺) = (𝑦𝐻𝐴) ↔ (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴)))
1413rspcev 3340 . . . . . . . 8 (((GId‘𝐺) ∈ 𝑋 ∧ (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴)) → ∃𝑦𝑋 (GId‘𝐺) = (𝑦𝐻𝐴))
158, 11, 14syl2anc 694 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑦𝑋 (GId‘𝐺) = (𝑦𝐻𝐴))
16 eqeq1 2655 . . . . . . . . 9 (𝑥 = (GId‘𝐺) → (𝑥 = (𝑦𝐻𝐴) ↔ (GId‘𝐺) = (𝑦𝐻𝐴)))
1716rexbidv 3081 . . . . . . . 8 (𝑥 = (GId‘𝐺) → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 (GId‘𝐺) = (𝑦𝐻𝐴)))
1817elrab 3396 . . . . . . 7 ((GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((GId‘𝐺) ∈ 𝑋 ∧ ∃𝑦𝑋 (GId‘𝐺) = (𝑦𝐻𝐴)))
198, 15, 18sylanbrc 699 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
20 eqeq1 2655 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝑢 = (𝑦𝐻𝐴)))
2120rexbidv 3081 . . . . . . . . . 10 (𝑥 = 𝑢 → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 𝑢 = (𝑦𝐻𝐴)))
22 oveq1 6697 . . . . . . . . . . . 12 (𝑦 = 𝑟 → (𝑦𝐻𝐴) = (𝑟𝐻𝐴))
2322eqeq2d 2661 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝑢 = (𝑦𝐻𝐴) ↔ 𝑢 = (𝑟𝐻𝐴)))
2423cbvrexv 3202 . . . . . . . . . 10 (∃𝑦𝑋 𝑢 = (𝑦𝐻𝐴) ↔ ∃𝑟𝑋 𝑢 = (𝑟𝐻𝐴))
2521, 24syl6bb 276 . . . . . . . . 9 (𝑥 = 𝑢 → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑟𝑋 𝑢 = (𝑟𝐻𝐴)))
2625elrab 3396 . . . . . . . 8 (𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑢𝑋 ∧ ∃𝑟𝑋 𝑢 = (𝑟𝐻𝐴)))
27 eqeq1 2655 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑣 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝑣 = (𝑦𝐻𝐴)))
2827rexbidv 3081 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑣 → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 𝑣 = (𝑦𝐻𝐴)))
29 oveq1 6697 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑠 → (𝑦𝐻𝐴) = (𝑠𝐻𝐴))
3029eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑠 → (𝑣 = (𝑦𝐻𝐴) ↔ 𝑣 = (𝑠𝐻𝐴)))
3130cbvrexv 3202 . . . . . . . . . . . . . . . 16 (∃𝑦𝑋 𝑣 = (𝑦𝐻𝐴) ↔ ∃𝑠𝑋 𝑣 = (𝑠𝐻𝐴))
3228, 31syl6bb 276 . . . . . . . . . . . . . . 15 (𝑥 = 𝑣 → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑠𝑋 𝑣 = (𝑠𝐻𝐴)))
3332elrab 3396 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑣𝑋 ∧ ∃𝑠𝑋 𝑣 = (𝑠𝐻𝐴)))
344, 9, 5rngodir 33834 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ (𝑟𝑋𝑠𝑋𝐴𝑋)) → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
35343exp2 1307 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ RingOps → (𝑟𝑋 → (𝑠𝑋 → (𝐴𝑋 → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴))))))
3635imp42 619 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝑟𝑋𝑠𝑋)) ∧ 𝐴𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
374, 5rngogcl 33841 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑟𝑋𝑠𝑋) → (𝑟𝐺𝑠) ∈ 𝑋)
38373expib 1287 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ RingOps → ((𝑟𝑋𝑠𝑋) → (𝑟𝐺𝑠) ∈ 𝑋))
3938imdistani 726 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ RingOps ∧ (𝑟𝑋𝑠𝑋)) → (𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋))
404, 9, 5rngocl 33830 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋𝐴𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋)
41403expa 1284 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋)
42 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)
43 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = (𝑟𝐺𝑠) → (𝑦𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴))
4443eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = (𝑟𝐺𝑠) → (((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴) ↔ ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)))
4544rspcev 3340 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑟𝐺𝑠) ∈ 𝑋 ∧ ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)) → ∃𝑦𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
4642, 45mpan2 707 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑟𝐺𝑠) ∈ 𝑋 → ∃𝑦𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
4746ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
48 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ((𝑟𝐺𝑠)𝐻𝐴) → (𝑥 = (𝑦𝐻𝐴) ↔ ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
4948rexbidv 3081 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ((𝑟𝐺𝑠)𝐻𝐴) → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
5049elrab 3396 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋 ∧ ∃𝑦𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
5141, 47, 50sylanbrc 699 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
5239, 51sylan 487 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝑟𝑋𝑠𝑋)) ∧ 𝐴𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
5336, 52eqeltrrd 2731 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ (𝑟𝑋𝑠𝑋)) ∧ 𝐴𝑋) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
5453an32s 863 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑟𝑋𝑠𝑋)) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
5554anassrs 681 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) ∧ 𝑠𝑋) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
56 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
5756eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑠𝐻𝐴) → (((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
5855, 57syl5ibrcom 237 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) ∧ 𝑠𝑋) → (𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
5958rexlimdva 3060 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → (∃𝑠𝑋 𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
6059adantld 482 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → ((𝑣𝑋 ∧ ∃𝑠𝑋 𝑣 = (𝑠𝐻𝐴)) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
6133, 60syl5bi 232 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → (𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
6261ralrimiv 2994 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → ∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
634, 9, 5rngoass 33835 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ (𝑤𝑋𝑟𝑋𝐴𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
64633exp2 1307 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ RingOps → (𝑤𝑋 → (𝑟𝑋 → (𝐴𝑋 → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴))))))
6564imp42 619 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ (𝑤𝑋𝑟𝑋)) ∧ 𝐴𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
6665an32s 863 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑤𝑋𝑟𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
674, 9, 5rngocl 33830 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ RingOps ∧ 𝑤𝑋𝑟𝑋) → (𝑤𝐻𝑟) ∈ 𝑋)
68673expib 1287 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ RingOps → ((𝑤𝑋𝑟𝑋) → (𝑤𝐻𝑟) ∈ 𝑋))
6968imdistani 726 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ (𝑤𝑋𝑟𝑋)) → (𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋))
704, 9, 5rngocl 33830 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋𝐴𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋)
71703expa 1284 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋)
72 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)
73 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑤𝐻𝑟) → (𝑦𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴))
7473eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑤𝐻𝑟) → (((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴) ↔ ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)))
7574rspcev 3340 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝐻𝑟) ∈ 𝑋 ∧ ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)) → ∃𝑦𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
7672, 75mpan2 707 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝐻𝑟) ∈ 𝑋 → ∃𝑦𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
7776ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
78 eqeq1 2655 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ((𝑤𝐻𝑟)𝐻𝐴) → (𝑥 = (𝑦𝐻𝐴) ↔ ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
7978rexbidv 3081 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑤𝐻𝑟)𝐻𝐴) → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
8079elrab 3396 . . . . . . . . . . . . . . . . . 18 (((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋 ∧ ∃𝑦𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
8171, 77, 80sylanbrc 699 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8269, 81sylan 487 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ (𝑤𝑋𝑟𝑋)) ∧ 𝐴𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8382an32s 863 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑤𝑋𝑟𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8466, 83eqeltrrd 2731 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑤𝑋𝑟𝑋)) → (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8584anass1rs 866 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) ∧ 𝑤𝑋) → (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8685ralrimiva 2995 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → ∀𝑤𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
8762, 86jca 553 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
88 oveq1 6697 . . . . . . . . . . . . . 14 (𝑢 = (𝑟𝐻𝐴) → (𝑢𝐺𝑣) = ((𝑟𝐻𝐴)𝐺𝑣))
8988eleq1d 2715 . . . . . . . . . . . . 13 (𝑢 = (𝑟𝐻𝐴) → ((𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
9089ralbidv 3015 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
91 oveq2 6698 . . . . . . . . . . . . . 14 (𝑢 = (𝑟𝐻𝐴) → (𝑤𝐻𝑢) = (𝑤𝐻(𝑟𝐻𝐴)))
9291eleq1d 2715 . . . . . . . . . . . . 13 (𝑢 = (𝑟𝐻𝐴) → ((𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
9392ralbidv 3015 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐻𝐴) → (∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ∀𝑤𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
9490, 93anbi12d 747 . . . . . . . . . . 11 (𝑢 = (𝑟𝐻𝐴) → ((∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}) ↔ (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
9587, 94syl5ibrcom 237 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑟𝑋) → (𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
9695rexlimdva 3060 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (∃𝑟𝑋 𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
9796adantld 482 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑢𝑋 ∧ ∃𝑟𝑋 𝑢 = (𝑟𝐻𝐴)) → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
9826, 97syl5bi 232 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} → (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
9998ralrimiv 2994 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∀𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))
1003, 19, 993jca 1261 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
1011, 100sylan 487 . . . 4 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})))
1024, 9, 5, 6isidlc 33944 . . . . 5 (𝑅 ∈ CRingOps → ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ↔ ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))))
103102adantr 480 . . . 4 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ↔ ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤𝑋 (𝑤𝐻𝑢) ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)}))))
104101, 103mpbird 247 . . 3 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅))
105 simpr 476 . . . . 5 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → 𝐴𝑋)
1064rneqi 5384 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
1075, 106eqtri 2673 . . . . . . . . 9 𝑋 = ran (1st𝑅)
108 eqid 2651 . . . . . . . . 9 (GId‘𝐻) = (GId‘𝐻)
109107, 9, 108rngo1cl 33868 . . . . . . . 8 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
110109adantr 480 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (GId‘𝐻) ∈ 𝑋)
1119, 107, 108rngolidm 33866 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((GId‘𝐻)𝐻𝐴) = 𝐴)
112111eqcomd 2657 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴 = ((GId‘𝐻)𝐻𝐴))
113 oveq1 6697 . . . . . . . . 9 (𝑦 = (GId‘𝐻) → (𝑦𝐻𝐴) = ((GId‘𝐻)𝐻𝐴))
114113eqeq2d 2661 . . . . . . . 8 (𝑦 = (GId‘𝐻) → (𝐴 = (𝑦𝐻𝐴) ↔ 𝐴 = ((GId‘𝐻)𝐻𝐴)))
115114rspcev 3340 . . . . . . 7 (((GId‘𝐻) ∈ 𝑋𝐴 = ((GId‘𝐻)𝐻𝐴)) → ∃𝑦𝑋 𝐴 = (𝑦𝐻𝐴))
116110, 112, 115syl2anc 694 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐻𝐴))
1171, 116sylan 487 . . . . 5 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐻𝐴))
118 eqeq1 2655 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝐴 = (𝑦𝐻𝐴)))
119118rexbidv 3081 . . . . . 6 (𝑥 = 𝐴 → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐻𝐴)))
120119elrab 3396 . . . . 5 (𝐴 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝐴𝑋 ∧ ∃𝑦𝑋 𝐴 = (𝑦𝐻𝐴)))
121105, 117, 120sylanbrc 699 . . . 4 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → 𝐴 ∈ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
122121snssd 4372 . . 3 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → {𝐴} ⊆ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
123 snssg 4347 . . . . . . . . 9 (𝐴𝑋 → (𝐴𝑗 ↔ {𝐴} ⊆ 𝑗))
124123biimpar 501 . . . . . . . 8 ((𝐴𝑋 ∧ {𝐴} ⊆ 𝑗) → 𝐴𝑗)
1254, 9, 5idllmulcl 33949 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ (𝐴𝑗𝑦𝑋)) → (𝑦𝐻𝐴) ∈ 𝑗)
126125anassrs 681 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) ∧ 𝑦𝑋) → (𝑦𝐻𝐴) ∈ 𝑗)
127 eleq1 2718 . . . . . . . . . . . . . 14 (𝑥 = (𝑦𝐻𝐴) → (𝑥𝑗 ↔ (𝑦𝐻𝐴) ∈ 𝑗))
128126, 127syl5ibrcom 237 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) ∧ 𝑦𝑋) → (𝑥 = (𝑦𝐻𝐴) → 𝑥𝑗))
129128rexlimdva 3060 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥𝑗))
130129adantr 480 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) ∧ 𝑥𝑋) → (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥𝑗))
131130ralrimiva 2995 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) → ∀𝑥𝑋 (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥𝑗))
132 rabss 3712 . . . . . . . . . 10 ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗 ↔ ∀𝑥𝑋 (∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥𝑗))
133131, 132sylibr 224 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑗) → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗)
134133ex 449 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝐴𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
135124, 134syl5 34 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → ((𝐴𝑋 ∧ {𝐴} ⊆ 𝑗) → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
136135expdimp 452 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴𝑋) → ({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
137136an32s 863 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑗 ∈ (Idl‘𝑅)) → ({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
138137ralrimiva 2995 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
1391, 138sylan 487 . . 3 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
140104, 122, 1393jca 1261 . 2 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗)))
141 snssi 4371 . . 3 (𝐴𝑋 → {𝐴} ⊆ 𝑋)
1424, 5igenval2 33995 . . 3 ((𝑅 ∈ RingOps ∧ {𝐴} ⊆ 𝑋) → ((𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))))
1431, 141, 142syl2an 493 . 2 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → ((𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ({𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))))
144140, 143mpbird 247 1 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607  {csn 4210  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  CRingOpsccring 33922  Idlcidl 33936   IdlGen cigen 33988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-com2 33919  df-crngo 33923  df-idl 33939  df-igen 33989
This theorem is referenced by:  isfldidl  33997  ispridlc  33999
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