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Theorem ig1pval 25690

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 3493	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = (Poly₁‘𝑅))
4		ig1pval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly₁‘𝑅)
5	3, 4	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = 𝑃)
6	5	fveq2d 6896	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = (LIdeal‘𝑃))
7		ig1pval.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2791	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = 𝑈)
9	5	fveq2d 6896	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = (0_g‘𝑃))
10		ig1pval.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑃)
11	9, 10	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = 0 )
12	11	sneqd 4641	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {(0_g‘(Poly₁‘𝑟))} = { 0 })
13	12	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 = {(0_g‘(Poly₁‘𝑟))} ↔ 𝑖 = { 0 }))
14		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = (Monic_1p‘𝑅))
15		ig1pval.m	. . . . . . . . . . 11 ⊢ 𝑀 = (Monic_1p‘𝑅)
16	14, 15	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = 𝑀)
17	16	ineq2d 4213	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (Monic_1p‘𝑟)) = (𝑖 ∩ 𝑀))
18		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = ( deg₁ ‘𝑅))
19		ig1pval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ( deg₁ ‘𝑅)
20	18, 19	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = 𝐷)
21	20	fveq1d 6894	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟)‘𝑔) = (𝐷‘𝑔))
22	12	difeq2d 4123	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))}) = (𝑖 ∖ { 0 }))
23	20, 22	imaeq12d 6061	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
24	23	infeq1d 9472	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
25	21, 24	eqeq12d 2749	. . . . . . . . 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 7368	. . . . . . . 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 4557	. . . . . . 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 5240	. . . . . 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 25652	. . . . . 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 7230	. . . . 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 2785	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝐺 = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
33	32	fveq1d 6894	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝐺‘𝐼) = ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34		eqeq1 2737	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35		ineq1 4206	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 ∩ 𝑀) = (𝐼 ∩ 𝑀))
36		difeq1 4116	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
37	36	imaeq2d 6060	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
38	37	infeq1d 9472	. . . . . 6 ⊢ (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
39	38	eqeq2d 2744	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
40	35, 39	riotaeqbidv 7368	. . . 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 4555	. . 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 2733	. . 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 6906	. . . 4 ⊢ 0 ∈ V
44		riotaex 7369	. . . 4 ⊢ (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
45	43, 44	ifex 4579	. . 3 ⊢ if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
46	41, 42, 45	fvmpt 6999	. 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 2793	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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ifcif 4529 {csn 4629 ↦ cmpt 5232 “ cima 5680 ‘cfv 6544 ℩crio 7364 infcinf 9436 ℝcr 11109 < clt 11248 0_gc0g 17385 LIdealclidl 20783 Poly₁cpl1 21701 deg₁ cdg1 25569 Monic_1pcmn1 25643 idlGen_1pcig1p 25647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-sup 9437 df-inf 9438 df-ig1p 25652

This theorem is referenced by: ig1pval2 25691 ig1pval3 25692

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