Theorem ig1pval 26549

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 = (idlGen1p‘𝑅)
ig1pval.z	⊢ 0 = (0g‘𝑃)
ig1pval.u	⊢ 𝑈 = (LIdeal‘𝑃)
ig1pval.d	⊢ 𝐷 = (deg1‘𝑅)
ig1pval.m	⊢ 𝑀 = (Monic1p‘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 = (idlGen1p‘𝑅)
2		elex 3475	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
4		ig1pval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
5	3, 4	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
6	5	fveq2d 6920	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
7		ig1pval.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
9	5	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
10		ig1pval.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑃)
11	9, 10	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
12	11	sneqd 4596	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {(0g‘(Poly1‘𝑟))} = { 0 })
13	12	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1‘𝑟))} ↔ 𝑖 = { 0 }))
14		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Monic1p‘𝑟) = (Monic1p‘𝑅))
15		ig1pval.m	. . . . . . . . . . 11 ⊢ 𝑀 = (Monic1p‘𝑅)
16	14, 15	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Monic1p‘𝑟) = 𝑀)
17	16	ineq2d 4120	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p‘𝑟)) = (𝑖 ∩ 𝑀))
18		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
19		ig1pval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (deg1‘𝑅)
20	18, 19	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
21	20	fveq1d 6918	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑔) = (𝐷‘𝑔))
22	12	difeq2d 4066	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1‘𝑟))}) = (𝑖 ∖ { 0 }))
23	20, 22	imaeq12d 6071	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
24	23	infeq1d 9507	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
25	21, 24	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
26	17, 25	riotaeqbidv 7411	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )) = (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
27	13, 11, 26	ifbieq12d 4514	. . . . . . 7 ⊢ (𝑟 = 𝑅 → if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ))) = if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))
28	8, 27	mpteq12dv 5198	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )))) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
29		df-ig1p 26508	. . . . . 6 ⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )))))
30	28, 29, 7	mptfvmpt 7265	. . . . 5 ⊢ (𝑅 ∈ V → (idlGen1p‘𝑅) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
31	2, 30	syl 18	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (idlGen1p‘𝑅) = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
32	1, 31	eqtrid 2810	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝐺 = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
33	32	fveq1d 6918	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝐺‘𝐼) = ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34		eqeq1 2767	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35		ineq1 4113	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 ∩ 𝑀) = (𝐼 ∩ 𝑀))
36		difeq1 4059	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
37	36	imaeq2d 6070	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
38	37	infeq1d 9507	. . . . . 6 ⊢ (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
39	38	eqeq2d 2774	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
40	35, 39	riotaeqbidv 7411	. . . 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 4512	. . 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 2763	. . 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 6930	. . . 4 ⊢ 0 ∈ V
44		riotaex 7412	. . . 4 ⊢ (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
45	43, 44	ifex 4536	. . 3 ⊢ if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
46	41, 42, 45	fvmpt 7024	. 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 2818	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092  ifcif 4485  {csn 4588   ↦ cmpt 5192   “ cima 5661  ‘cfv 6566  ℩crio 7407  infcinf 9470  ℝcr 11238   < clt 11382  0_gc0g 17660  LIdealclidl 21535  Poly₁cpl1 22550  deg₁cdg1 26427  Monic_1pcmn1 26499  idlGen_1pcig1p 26503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-sup 9471  df-inf 9472  df-ig1p 26508

This theorem is used by:  ig1pval2  26550  ig1pval3  26551