algextdeglem7 - Mathbox for Thierry Arnoux

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Theorem algextdeglem7 33083

Description: Lemma for algextdeg 33085. (Contributed by Thierry Arnoux, 2-Apr-2025.)

**Hypotheses**
Ref	Expression
algextdeg.k	⊢ 𝐾 = (𝐸 ↾_s 𝐹)
algextdeg.l	⊢ 𝐿 = (𝐸 ↾_s (𝐸 fldGen (𝐹 ∪ {𝐴})))
algextdeg.d	⊢ 𝐷 = ( deg₁ ‘𝐸)
algextdeg.m	⊢ 𝑀 = (𝐸 minPoly 𝐹)
algextdeg.f	⊢ (𝜑 → 𝐸 ∈ Field)
algextdeg.e	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
algextdeg.a	⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
algextdeglem.o	⊢ 𝑂 = (𝐸 evalSub₁ 𝐹)
algextdeglem.y	⊢ 𝑃 = (Poly₁‘𝐾)
algextdeglem.u	⊢ 𝑈 = (Base‘𝑃)
algextdeglem.g	⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
algextdeglem.n	⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~_QG 𝑍))
algextdeglem.z	⊢ 𝑍 = (^◡𝐺 “ {(0_g‘𝐿)})
algextdeglem.q	⊢ 𝑄 = (𝑃 /_s (𝑃 ~_QG 𝑍))
algextdeglem.j	⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))
algextdeglem.r	⊢ 𝑅 = (rem_1p‘𝐾)
algextdeglem.h	⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
algextdeglem.t	⊢ 𝑇 = (^◡( deg₁ ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))
algextdeglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)

**Assertion**
Ref	Expression
algextdeglem7	⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋))

Distinct variable groups: 𝐴,𝑝 𝐸,𝑝 𝐹,𝑝 𝑀,𝑝 𝑂,𝑝 𝑃,𝑝 𝑅,𝑝 𝑈,𝑝 𝑋,𝑝 𝜑,𝑝 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑥) 𝐷(𝑥,𝑝) 𝑃(𝑥) 𝑄(𝑥,𝑝) 𝑅(𝑥) 𝑇(𝑥,𝑝) 𝑈(𝑥) 𝐸(𝑥) 𝐹(𝑥) 𝐺(𝑥,𝑝) 𝐻(𝑥,𝑝) 𝐽(𝑥,𝑝) 𝐾(𝑥,𝑝) 𝐿(𝑥,𝑝) 𝑀(𝑥) 𝑁(𝑥,𝑝) 𝑂(𝑥) 𝑋(𝑥) 𝑍(𝑥,𝑝)

**Proof of Theorem algextdeglem7**
Step	Hyp	Ref	Expression
1		algextdeg.k	. . . . 5 ⊢ 𝐾 = (𝐸 ↾_s 𝐹)
2		algextdeg.d	. . . . 5 ⊢ 𝐷 = ( deg₁ ‘𝐸)
3		algextdeglem.y	. . . . 5 ⊢ 𝑃 = (Poly₁‘𝐾)
4		algextdeglem.u	. . . . 5 ⊢ 𝑈 = (Base‘𝑃)
5		algextdeglem.o	. . . . . . 7 ⊢ 𝑂 = (𝐸 evalSub₁ 𝐹)
6	1	fveq2i 6894	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘(𝐸 ↾_s 𝐹))
7	3, 6	eqtri 2759	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘(𝐸 ↾_s 𝐹))
8		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		algextdeg.f	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
10		algextdeg.e	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
11		eqid 2731	. . . . . . . . 9 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
12	9	fldcrngd 20517	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
13		sdrgsubrg 20554	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
14	10, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
15	5, 1, 8, 11, 12, 14	irngssv 33056	. . . . . . . 8 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		algextdeg.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 3983	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2731	. . . . . . 7 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0_g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0_g‘𝐸)}
19		eqid 2731	. . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
20		eqid 2731	. . . . . . 7 ⊢ (idlGen_1p‘(𝐸 ↾_s 𝐹)) = (idlGen_1p‘(𝐸 ↾_s 𝐹))
21		algextdeg.m	. . . . . . 7 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
22	5, 7, 8, 9, 10, 17, 11, 18, 19, 20, 21	minplycl 33071	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
23	22, 4	eleqtrrdi 2843	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
24	1, 2, 3, 4, 23, 14	ressdeg1 32940	. . . 4 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = (( deg₁ ‘𝐾)‘(𝑀‘𝐴)))
25	24	breq2d 5160	. . 3 ⊢ (𝜑 → ((( deg₁ ‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴)) ↔ (( deg₁ ‘𝐾)‘𝑋) < (( deg₁ ‘𝐾)‘(𝑀‘𝐴))))
26		algextdeglem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
27		eqid 2731	. . . . 5 ⊢ ( deg₁ ‘𝐾) = ( deg₁ ‘𝐾)
28		algextdeglem.t	. . . . 5 ⊢ 𝑇 = (^◡( deg₁ ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))
29	9	flddrngd 20516	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
30	29	drngringd 20512	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Ring)
31		eqid 2731	. . . . . . . . 9 ⊢ (Poly₁‘𝐸) = (Poly₁‘𝐸)
32		eqid 2731	. . . . . . . . 9 ⊢ (PwSer₁‘𝐾) = (PwSer₁‘𝐾)
33		eqid 2731	. . . . . . . . 9 ⊢ (Base‘(PwSer₁‘𝐾)) = (Base‘(PwSer₁‘𝐾))
34		eqid 2731	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝐸)) = (Base‘(Poly₁‘𝐸))
35	31, 1, 3, 4, 14, 32, 33, 34	ressply1bas2 21983	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer₁‘𝐾)) ∩ (Base‘(Poly₁‘𝐸))))
36		inss2 4229	. . . . . . . 8 ⊢ ((Base‘(PwSer₁‘𝐾)) ∩ (Base‘(Poly₁‘𝐸))) ⊆ (Base‘(Poly₁‘𝐸))
37	35, 36	eqsstrdi 4036	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly₁‘𝐸)))
38	37, 23	sseldd 3983	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly₁‘𝐸)))
39		eqid 2731	. . . . . . 7 ⊢ (0_g‘(Poly₁‘𝐸)) = (0_g‘(Poly₁‘𝐸))
40	39, 9, 10, 21, 16	irngnminplynz 33075	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0_g‘(Poly₁‘𝐸)))
41	2, 31, 39, 34	deg1nn0cl 25855	. . . . . 6 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly₁‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0_g‘(Poly₁‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ₀)
42	30, 38, 40, 41	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ₀)
43		fldsdrgfld 20561	. . . . . . . . 9 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾_s 𝐹) ∈ Field)
44	9, 10, 43	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Field)
45	1, 44	eqeltrid 2836	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field)
46		fldidom 21127	. . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
47	45, 46	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn)
48	47	idomringd 32660	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
49	3, 27, 28, 42, 48, 4	ply1degleel 32956	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝑋 ∈ 𝑈 ∧ (( deg₁ ‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴)))))
50	26, 49	mpbirand 704	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (( deg₁ ‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴))))
51		eqid 2731	. . . 4 ⊢ (Unic_1p‘𝐾) = (Unic_1p‘𝐾)
52		algextdeglem.r	. . . 4 ⊢ 𝑅 = (rem_1p‘𝐾)
53	1	fveq2i 6894	. . . . . 6 ⊢ (Monic_1p‘𝐾) = (Monic_1p‘(𝐸 ↾_s 𝐹))
54	39, 9, 10, 21, 16, 53	minplym1p 33076	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic_1p‘𝐾))
55		eqid 2731	. . . . . 6 ⊢ (Monic_1p‘𝐾) = (Monic_1p‘𝐾)
56	51, 55	mon1puc1p 25917	. . . . 5 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic_1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic_1p‘𝐾))
57	48, 54, 56	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic_1p‘𝐾))
58	3, 4, 51, 52, 47, 27, 26, 57	r1pid2 32969	. . 3 ⊢ (𝜑 → ((𝑋𝑅(𝑀‘𝐴)) = 𝑋 ↔ (( deg₁ ‘𝐾)‘𝑋) < (( deg₁ ‘𝐾)‘(𝑀‘𝐴))))
59	25, 50, 58	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝑋𝑅(𝑀‘𝐴)) = 𝑋))
60		algextdeglem.h	. . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
61		oveq1 7419	. . . 4 ⊢ (𝑝 = 𝑋 → (𝑝𝑅(𝑀‘𝐴)) = (𝑋𝑅(𝑀‘𝐴)))
62		ovexd 7447	. . . 4 ⊢ (𝜑 → (𝑋𝑅(𝑀‘𝐴)) ∈ V)
63	60, 61, 26, 62	fvmptd3 7021	. . 3 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑋𝑅(𝑀‘𝐴)))
64	63	eqeq1d 2733	. 2 ⊢ (𝜑 → ((𝐻‘𝑋) = 𝑋 ↔ (𝑋𝑅(𝑀‘𝐴)) = 𝑋))
65	59, 64	bitr4d 282	1 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 Vcvv 3473 ∪ cun 3946 ∩ cin 3947 {csn 4628 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 dom cdm 5676 “ cima 5679 ‘cfv 6543 (class class class)co 7412 [cec 8707 -∞cmnf 11253 < clt 11255 ℕ₀cn0 12479 [,)cico 13333 Basecbs 17151 ↾_s cress 17180 0_gc0g 17392 /_s cqus 17458 ~_QG cqg 19042 Ringcrg 20131 SubRingcsubrg 20461 Fieldcfield 20505 SubDRingcsdrg 20549 RSpancrsp 20933 IDomncidom 21101 PwSer₁cps1 21931 Poly₁cpl1 21933 evalSub₁ ces1 22065 deg₁ cdg1 25818 Monic_1pcmn1 25892 Unic_1pcuc1p 25893 rem_1pcr1p 25895 idlGen_1pcig1p 25896 fldGen cfldgen 32685 IntgRing cirng 33051 minPoly cminply 33060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-ico 13337 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-cntz 19226 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-srg 20085 df-ring 20133 df-cring 20134 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-rhm 20367 df-nzr 20408 df-subrng 20438 df-subrg 20463 df-drng 20506 df-field 20507 df-sdrg 20550 df-lmod 20620 df-lss 20691 df-lsp 20731 df-sra 20934 df-rgmod 20935 df-lidl 20936 df-rsp 20937 df-rlreg 21103 df-domn 21104 df-idom 21105 df-cnfld 21149 df-assa 21631 df-asp 21632 df-ascl 21633 df-psr 21685 df-mvr 21686 df-mpl 21687 df-opsr 21689 df-evls 21859 df-evl 21860 df-psr1 21936 df-vr1 21937 df-ply1 21938 df-coe1 21939 df-evls1 22067 df-evl1 22068 df-mdeg 25819 df-deg1 25820 df-mon1 25897 df-uc1p 25898 df-q1p 25899 df-r1p 25900 df-ig1p 25901 df-irng 33052 df-minply 33061

This theorem is referenced by: algextdeglem8 33084

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