Theorem ig1pval 26487

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 3472	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
4		ig1pval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
5	3, 4	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
6	5	fveq2d 6887	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
7		ig1pval.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2814	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
9	5	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
10		ig1pval.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑃)
11	9, 10	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
12	11	sneqd 4596	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {(0g‘(Poly1‘𝑟))} = { 0 })
13	12	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1‘𝑟))} ↔ 𝑖 = { 0 }))
14		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Monic1p‘𝑟) = (Monic1p‘𝑅))
15		ig1pval.m	. . . . . . . . . . 11 ⊢ 𝑀 = (Monic1p‘𝑅)
16	14, 15	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Monic1p‘𝑟) = 𝑀)
17	16	ineq2d 4166	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p‘𝑟)) = (𝑖 ∩ 𝑀))
18		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
19		ig1pval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (deg1‘𝑅)
20	18, 19	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
21	20	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑔) = (𝐷‘𝑔))
22	12	difeq2d 4074	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1‘𝑟))}) = (𝑖 ∖ { 0 }))
23	20, 22	imaeq12d 6053	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
24	23	infeq1d 9463	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
25	21, 24	eqeq12d 2777	. . . . . . . . 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 7378	. . . . . . . 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 4511	. . . . . . 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 5192	. . . . . 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 26446	. . . . . 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 7232	. . . . 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 2808	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝐺 = (𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
33	32	fveq1d 6885	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝐺‘𝐼) = ((𝑖 ∈ 𝑈 ↦ if(𝑖 = { 0 }, 0 , (℩𝑔 ∈ (𝑖 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34		eqeq1 2765	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35		ineq1 4159	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 ∩ 𝑀) = (𝐼 ∩ 𝑀))
36		difeq1 4067	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
37	36	imaeq2d 6052	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
38	37	infeq1d 9463	. . . . . 6 ⊢ (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
39	38	eqeq2d 2772	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐷‘𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
40	35, 39	riotaeqbidv 7378	. . . 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 4509	. . 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 2761	. . 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 6897	. . . 4 ⊢ 0 ∈ V
44		riotaex 7379	. . . 4 ⊢ (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
45	43, 44	ifex 4533	. . 3 ⊢ if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
46	41, 42, 45	fvmpt 6991	. 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 2816	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 2145  Vcvv 3451   ∖ cdif 3896   ∩ cin 3898  ifcif 4482  {csn 4584   ↦ cmpt 5186   “ cima 5654  ‘cfv 6537  ℩crio 7374  infcinf 9426  ℝcr 11192   < clt 11336  0_gc0g 17603  LIdealclidl 21477  Poly₁cpl1 22488  deg₁cdg1 26365  Monic_1pcmn1 26437  idlGen_1pcig1p 26441

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-sup 9427  df-inf 9428  df-ig1p 26446

This theorem is used by:  ig1pval2  26488  ig1pval3  26489