![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isalgnb | Structured version Visualization version GIF version |
Description: Property for an element 𝑋 of a field 𝐸 to be algebraic over a subfield 𝐹. (Contributed by Thierry Arnoux, 26-Jan-2025.) |
Ref | Expression |
---|---|
algnbval.o | ⊢ 𝑂 = (𝐸 evalSub1 𝐹) |
algnbval.z | ⊢ 𝑍 = (0g‘(Poly1‘𝐸)) |
algnbval.1 | ⊢ 0 = (0g‘𝐸) |
algnbval.2 | ⊢ (𝜑 → 𝐸 ∈ Field) |
algnbval.3 | ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) |
isalgnb.b | ⊢ 𝐵 = (Base‘𝐸) |
Ref | Expression |
---|---|
isalgnb | ⊢ (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂‘𝑝)‘𝑋) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algnbval.o | . . . . . 6 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
2 | algnbval.z | . . . . . 6 ⊢ 𝑍 = (0g‘(Poly1‘𝐸)) | |
3 | algnbval.1 | . . . . . 6 ⊢ 0 = (0g‘𝐸) | |
4 | algnbval.2 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Field) | |
5 | algnbval.3 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
6 | 1, 2, 3, 4, 5 | algnbval 32174 | . . . . 5 ⊢ (𝜑 → (𝐸 AlgNb 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) |
7 | 6 | eleq2d 2823 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ 𝑋 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))) |
8 | eliun 4956 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑋 ∈ (◡(𝑂‘𝑝) “ { 0 })) | |
9 | 7, 8 | bitrdi 286 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑋 ∈ (◡(𝑂‘𝑝) “ { 0 }))) |
10 | eqid 2737 | . . . . . 6 ⊢ (𝐸 ↑s 𝐵) = (𝐸 ↑s 𝐵) | |
11 | isalgnb.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐸) | |
12 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝐸 ↑s 𝐵)) = (Base‘(𝐸 ↑s 𝐵)) | |
13 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐸 ∈ Field) |
14 | 11 | fvexi 6853 | . . . . . . 7 ⊢ 𝐵 ∈ V |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐵 ∈ V) |
16 | 4 | fldcrngd 20149 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ CRing) |
17 | issdrg 20213 | . . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
18 | 17 | simp2bi 1146 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) |
19 | 5, 18 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
20 | eqid 2737 | . . . . . . . . . . 11 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹) | |
21 | eqid 2737 | . . . . . . . . . . 11 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹)) | |
22 | 1, 11, 10, 20, 21 | evls1rhm 21639 | . . . . . . . . . 10 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s 𝐵))) |
23 | 16, 19, 22 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s 𝐵))) |
24 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))) | |
25 | 24, 12 | rhmf 20110 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s 𝐵)) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s 𝐵))) |
26 | 23, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s 𝐵))) |
27 | 26 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s 𝐵))) |
28 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) | |
29 | 28 | eldifad 3920 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ dom 𝑂) |
30 | 26 | fdmd 6676 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))) |
31 | 30 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))) |
32 | 29, 31 | eleqtrd 2840 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹)))) |
33 | 27, 32 | ffvelcdmd 7032 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s 𝐵))) |
34 | 10, 11, 12, 13, 15, 33 | pwselbas 17330 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):𝐵⟶𝐵) |
35 | ffn 6665 | . . . . 5 ⊢ ((𝑂‘𝑝):𝐵⟶𝐵 → (𝑂‘𝑝) Fn 𝐵) | |
36 | fniniseg 7007 | . . . . 5 ⊢ ((𝑂‘𝑝) Fn 𝐵 → (𝑋 ∈ (◡(𝑂‘𝑝) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑝)‘𝑋) = 0 ))) | |
37 | 34, 35, 36 | 3syl 18 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑋 ∈ (◡(𝑂‘𝑝) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑝)‘𝑋) = 0 ))) |
38 | 37 | rexbidva 3171 | . . 3 ⊢ (𝜑 → (∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑋 ∈ (◡(𝑂‘𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑝)‘𝑋) = 0 ))) |
39 | 9, 38 | bitrd 278 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑝)‘𝑋) = 0 ))) |
40 | r19.42v 3185 | . 2 ⊢ (∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑝)‘𝑋) = 0 ) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂‘𝑝)‘𝑋) = 0 )) | |
41 | 39, 40 | bitrdi 286 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂‘𝑝)‘𝑋) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 Vcvv 3443 ∖ cdif 3905 {csn 4584 ∪ ciun 4952 ◡ccnv 5630 dom cdm 5631 “ cima 5634 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 ↾s cress 17071 0gc0g 17280 ↑s cpws 17287 CRingccrg 19918 RingHom crh 20095 DivRingcdr 20137 Fieldcfield 20138 SubRingcsubrg 20170 SubDRingcsdrg 20211 Poly1cpl1 21499 evalSub1 ces1 21630 AlgNb calgnb 32170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14184 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-hom 17116 df-cco 17117 df-0g 17282 df-gsum 17283 df-prds 17288 df-pws 17290 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-submnd 18561 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-ghm 18964 df-cntz 19055 df-cmn 19522 df-abl 19523 df-mgp 19855 df-ur 19872 df-srg 19876 df-ring 19919 df-cring 19920 df-rnghom 20098 df-field 20140 df-subrg 20172 df-sdrg 20212 df-lmod 20276 df-lss 20345 df-lsp 20385 df-assa 21211 df-asp 21212 df-ascl 21213 df-psr 21263 df-mvr 21264 df-mpl 21265 df-opsr 21267 df-evls 21433 df-psr1 21502 df-ply1 21504 df-evls1 21632 df-algnb 32172 |
This theorem is referenced by: minplyeulem 32176 |
Copyright terms: Public domain | W3C validator |