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Theorem ig1pval 25689

Description: Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)

**Hypotheses**
Ref	Expression
ig1pval.p	⊢ 𝑃 = (Poly₁‘𝑅)
ig1pval.g	⊢ 𝐺 = (idlGen_1p‘𝑅)
ig1pval.z	⊢ 0 = (0_g‘𝑃)
ig1pval.u	⊢ 𝑈 = (LIdeal‘𝑃)
ig1pval.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
ig1pval.m	⊢ 𝑀 = (Monic_1p‘𝑅)

**Assertion**
Ref	Expression
ig1pval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))

Distinct variable groups: 𝑔,𝐼 𝑔,𝑀 𝑅,𝑔 Allowed substitution hints: 𝐷(𝑔) 𝑃(𝑔) 𝑈(𝑔) 𝐺(𝑔) 𝑉(𝑔) 0 (𝑔)

Proof of Theorem ig1pval Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ig1pval.g	. . . 4 ⊢ 𝐺 = (idlGen_1p‘𝑅)
2		elex 3492	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = (Poly₁‘𝑅))
4		ig1pval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly₁‘𝑅)
5	3, 4	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = 𝑃)
6	5	fveq2d 6895	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = (LIdeal‘𝑃))
7		ig1pval.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2790	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = 𝑈)
9	5	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = (0_g‘𝑃))
10		ig1pval.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑃)
11	9, 10	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = 0 )
12	11	sneqd 4640	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {(0_g‘(Poly₁‘𝑟))} = { 0 })
13	12	eqeq2d 2743	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 = {(0_g‘(Poly₁‘𝑟))} ↔ 𝑖 = { 0 }))
14		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = (Monic_1p‘𝑅))
15		ig1pval.m	. . . . . . . . . . 11 ⊢ 𝑀 = (Monic_1p‘𝑅)
16	14, 15	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = 𝑀)
17	16	ineq2d 4212	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (Monic_1p‘𝑟)) = (𝑖 ∩ 𝑀))
18		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = ( deg₁ ‘𝑅))
19		ig1pval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ( deg₁ ‘𝑅)
20	18, 19	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = 𝐷)
21	20	fveq1d 6893	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟)‘𝑔) = (𝐷‘𝑔))
22	12	difeq2d 4122	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))}) = (𝑖 ∖ { 0 }))
23	20, 22	imaeq12d 6060	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
24	23	infeq1d 9471	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
25	21, 24	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
26	17, 25	riotaeqbidv 7367	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))(( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < )) = (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
27	13, 11, 26	ifbieq12d 4556	. . . . . . 7 ⊢ (𝑟 = 𝑅 → if(𝑖 = {(0_g‘(Poly₁‘𝑟))}, (0_g‘(Poly₁‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))(( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < ))) = if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))
28	8, 27	mpteq12dv 5239	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ if(𝑖 = {(0_g‘(Poly₁‘𝑟))}, (0_g‘(Poly₁‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))(( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < )))) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
29		df-ig1p 25651	. . . . . 6 ⊢ idlGen_1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ if(𝑖 = {(0_g‘(Poly₁‘𝑟))}, (0_g‘(Poly₁‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))(( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < )))))
30	28, 29, 7	mptfvmpt 7229	. . . . 5 ⊢ (𝑅 ∈ V → (idlGen_1p‘𝑅) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
31	2, 30	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (idlGen_1p‘𝑅) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
32	1, 31	eqtrid 2784	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝐺 = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
33	32	fveq1d 6893	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝐺‘𝐼) = ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34		eqeq1 2736	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35		ineq1 4205	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 ∩ 𝑀) = (𝐼 ∩ 𝑀))
36		difeq1 4115	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
37	36	imaeq2d 6059	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
38	37	infeq1d 9471	. . . . . 6 ⊢ (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
39	38	eqeq2d 2743	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
40	35, 39	riotaeqbidv 7367	. . . 4 ⊢ (𝑖 = 𝐼 → (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )) = (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
41	34, 40	ifbieq2d 4554	. . 3 ⊢ (𝑖 = 𝐼 → if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
42		eqid 2732	. . 3 ⊢ (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))
43	10	fvexi 6905	. . . 4 ⊢ 0 ∈ V
44		riotaex 7368	. . . 4 ⊢ (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
45	43, 44	ifex 4578	. . 3 ⊢ if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
46	41, 42, 45	fvmpt 6998	. 2 ⊢ (𝐼 ∈ 𝑈 → ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
47	33, 46	sylan9eq 2792	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3945 ∩ cin 3947 ifcif 4528 {csn 4628 ↦ cmpt 5231 “ cima 5679 ‘cfv 6543 ℩crio 7363 infcinf 9435 ℝcr 11108 < clt 11247 0_gc0g 17384 LIdealclidl 20782 Poly₁cpl1 21700 deg₁ cdg1 25568 Monic_1pcmn1 25642 idlGen_1pcig1p 25646

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-sup 9436 df-inf 9437 df-ig1p 25651

This theorem is referenced by: ig1pval2 25690 ig1pval3 25691

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