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Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem5 | Structured version Visualization version GIF version |
Description: Lemma for algextdeg 33731. The subspace 𝑍 of annihilators of 𝐴 is a principal ideal generated by the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
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 “ {(0g‘𝐿)}) |
algextdeglem.q | ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍)) |
algextdeglem.j | ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝)) |
Ref | Expression |
---|---|
algextdeglem5 | ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algextdeglem.o | . . 3 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
2 | algextdeglem.y | . . . 4 ⊢ 𝑃 = (Poly1‘𝐾) | |
3 | algextdeg.k | . . . . 5 ⊢ 𝐾 = (𝐸 ↾s 𝐹) | |
4 | 3 | fveq2i 6910 | . . . 4 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹)) |
5 | 2, 4 | eqtri 2763 | . . 3 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) |
6 | eqid 2735 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
7 | algextdeg.f | . . 3 ⊢ (𝜑 → 𝐸 ∈ Field) | |
8 | algextdeg.e | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
9 | eqid 2735 | . . . . 5 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
10 | 7 | fldcrngd 20759 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing) |
11 | issdrg 20806 | . . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
12 | 8, 11 | sylib 218 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
13 | 12 | simp2d 1142 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
14 | 1, 3, 6, 9, 10, 13 | irngssv 33703 | . . . 4 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
15 | algextdeg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
16 | 14, 15 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
17 | eqid 2735 | . . 3 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} | |
18 | eqid 2735 | . . 3 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
19 | eqid 2735 | . . 3 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
20 | 1, 5, 6, 7, 8, 16, 9, 17, 18, 19 | ply1annig1p 33712 | . 2 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = ((RSpan‘𝑃)‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})) |
21 | algextdeglem.z | . . . 4 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)}) | |
22 | 10 | crnggrpd 20265 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Grp) |
23 | 22 | grpmndd 18977 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Mnd) |
24 | 7 | flddrngd 20758 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
25 | subrgsubg 20594 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸)) | |
26 | 6 | subgss 19158 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸)) |
27 | 13, 25, 26 | 3syl 18 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸)) |
28 | 16 | snssd 4814 | . . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸)) |
29 | 27, 28 | unssd 4202 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸)) |
30 | 6, 24, 29 | fldgensdrg 33296 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸)) |
31 | sdrgsubrg 20809 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸)) | |
32 | subrgsubg 20594 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸)) | |
33 | 9 | subg0cl 19165 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
34 | 30, 31, 32, 33 | 4syl 19 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
35 | 6, 24, 29 | fldgenssv 33297 | . . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) |
36 | algextdeg.l | . . . . . . . 8 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) | |
37 | 36, 6, 9 | ress0g 18788 | . . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) → (0g‘𝐸) = (0g‘𝐿)) |
38 | 23, 34, 35, 37 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐿)) |
39 | 38 | sneqd 4643 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐸)} = {(0g‘𝐿)}) |
40 | 39 | imaeq2d 6080 | . . . 4 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)})) |
41 | 21, 40 | eqtr4id 2794 | . . 3 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)})) |
42 | algextdeglem.g | . . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) | |
43 | algextdeglem.u | . . . . . 6 ⊢ 𝑈 = (Base‘𝑃) | |
44 | 43 | mpteq1i 5244 | . . . . 5 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴)) |
45 | 42, 44 | eqtri 2763 | . . . 4 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴)) |
46 | 1, 5, 6, 10, 13, 16, 9, 17, 45 | ply1annidllem 33709 | . . 3 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)})) |
47 | 41, 46 | eqtr4d 2778 | . 2 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) |
48 | algextdeg.m | . . . . 5 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
49 | 1, 5, 6, 7, 8, 16, 9, 17, 18, 19, 48 | minplyval 33713 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})) |
50 | 49 | sneqd 4643 | . . 3 ⊢ (𝜑 → {(𝑀‘𝐴)} = {((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})}) |
51 | 50 | fveq2d 6911 | . 2 ⊢ (𝜑 → ((RSpan‘𝑃)‘{(𝑀‘𝐴)}) = ((RSpan‘𝑃)‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})) |
52 | 20, 47, 51 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {crab 3433 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 “ cima 5692 ‘cfv 6563 (class class class)co 7431 [cec 8742 Basecbs 17245 ↾s cress 17274 0gc0g 17486 /s cqus 17552 Mndcmnd 18760 SubGrpcsubg 19151 ~QG cqg 19153 SubRingcsubrg 20586 DivRingcdr 20746 Fieldcfield 20747 SubDRingcsdrg 20804 RSpancrsp 21235 Poly1cpl1 22194 evalSub1 ces1 22333 deg1cdg1 26108 idlGen1pcig1p 26184 fldGen cfldgen 33292 IntgRing cirng 33698 minPoly cminply 33707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-drng 20748 df-field 20749 df-sdrg 20805 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-cnfld 21383 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 df-ig1p 26189 df-fldgen 33293 df-irng 33699 df-minply 33708 |
This theorem is referenced by: algextdeglem6 33728 |
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