ig1pval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ig1pval		Structured version Visualization version GIF version

Theorem ig1pval 25925

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 3491	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = (Poly₁‘𝑅))
4		ig1pval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly₁‘𝑅)
5	3, 4	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly₁‘𝑟) = 𝑃)
6	5	fveq2d 6894	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = (LIdeal‘𝑃))
7		ig1pval.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2788	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly₁‘𝑟)) = 𝑈)
9	5	fveq2d 6894	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = (0_g‘𝑃))
10		ig1pval.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑃)
11	9, 10	eqtr4di 2788	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0_g‘(Poly₁‘𝑟)) = 0 )
12	11	sneqd 4639	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {(0_g‘(Poly₁‘𝑟))} = { 0 })
13	12	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 = {(0_g‘(Poly₁‘𝑟))} ↔ 𝑖 = { 0 }))
14		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = (Monic_1p‘𝑅))
15		ig1pval.m	. . . . . . . . . . 11 ⊢ 𝑀 = (Monic_1p‘𝑅)
16	14, 15	eqtr4di 2788	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Monic_1p‘𝑟) = 𝑀)
17	16	ineq2d 4211	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (Monic_1p‘𝑟)) = (𝑖 ∩ 𝑀))
18		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = ( deg₁ ‘𝑅))
19		ig1pval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ( deg₁ ‘𝑅)
20	18, 19	eqtr4di 2788	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ( deg₁ ‘𝑟) = 𝐷)
21	20	fveq1d 6892	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟)‘𝑔) = (𝐷‘𝑔))
22	12	difeq2d 4121	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))}) = (𝑖 ∖ { 0 }))
23	20, 22	imaeq12d 6059	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
24	23	infeq1d 9474	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
25	21, 24	eqeq12d 2746	. . . . . . . . 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 7370	. . . . . . . 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 4555	. . . . . . 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 5238	. . . . . 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 25887	. . . . . 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 7231	. . . . 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 2782	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝐺 = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
33	32	fveq1d 6892	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝐺‘𝐼) = ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34		eqeq1 2734	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35		ineq1 4204	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 ∩ 𝑀) = (𝐼 ∩ 𝑀))
36		difeq1 4114	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
37	36	imaeq2d 6058	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
38	37	infeq1d 9474	. . . . . 6 ⊢ (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
39	38	eqeq2d 2741	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
40	35, 39	riotaeqbidv 7370	. . . 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 4553	. . 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 2730	. . 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 6904	. . . 4 ⊢ 0 ∈ V
44		riotaex 7371	. . . 4 ⊢ (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
45	43, 44	ifex 4577	. . 3 ⊢ if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
46	41, 42, 45	fvmpt 6997	. 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 2790	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 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∖ cdif 3944 ∩ cin 3946 ifcif 4527 {csn 4627 ↦ cmpt 5230 “ cima 5678 ‘cfv 6542 ℩crio 7366 infcinf 9438 ℝcr 11111 < clt 11252 0_gc0g 17389 LIdealclidl 20928 Poly₁cpl1 21920 deg₁ cdg1 25804 Monic_1pcmn1 25878 idlGen_1pcig1p 25882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-sup 9439 df-inf 9440 df-ig1p 25887

This theorem is referenced by: ig1pval2 25926 ig1pval3 25927

W3C validator