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Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem1 | Structured version Visualization version GIF version |
Description: Lemma for algextdeg 33731. (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)) |
Ref | Expression |
---|---|
algextdeglem1 | ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algextdeg.l | . . . 4 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) | |
2 | 1 | oveq1i 7441 | . . 3 ⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s 𝐹) |
3 | ovex 7464 | . . . 4 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V | |
4 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
5 | algextdeg.e | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
6 | issdrg 20806 | . . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
7 | 5, 6 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
8 | 7 | simp1d 1141 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
9 | 7 | simp2d 1142 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
10 | subrgsubg 20594 | . . . . . . . 8 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸)) | |
11 | 4 | subgss 19158 | . . . . . . . 8 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸)) |
12 | 9, 10, 11 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸)) |
13 | eqid 2735 | . . . . . . . . . 10 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹) | |
14 | algextdeg.k | . . . . . . . . . 10 ⊢ 𝐾 = (𝐸 ↾s 𝐹) | |
15 | eqid 2735 | . . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
16 | algextdeg.f | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Field) | |
17 | 16 | fldcrngd 20759 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ CRing) |
18 | 13, 14, 4, 15, 17, 9 | irngssv 33703 | . . . . . . . . 9 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
19 | algextdeg.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
20 | 18, 19 | sseldd 3996 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
21 | 20 | snssd 4814 | . . . . . . 7 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸)) |
22 | 12, 21 | unssd 4202 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸)) |
23 | 4, 8, 22 | fldgenssid 33295 | . . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
24 | 23 | unssad 4203 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
25 | ressabs 17295 | . . . 4 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s 𝐹) = (𝐸 ↾s 𝐹)) | |
26 | 3, 24, 25 | sylancr 587 | . . 3 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s 𝐹) = (𝐸 ↾s 𝐹)) |
27 | 2, 26 | eqtrid 2787 | . 2 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹)) |
28 | 27, 14 | eqtr4di 2793 | 1 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 0gc0g 17486 SubGrpcsubg 19151 SubRingcsubrg 20586 DivRingcdr 20746 Fieldcfield 20747 SubDRingcsdrg 20804 evalSub1 ces1 22333 deg1cdg1 26108 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 |
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-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-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-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 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-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-sdrg 20805 df-lmod 20877 df-lss 20948 df-lsp 20988 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-psr1 22197 df-ply1 22199 df-evls1 22335 df-mon1 26185 df-fldgen 33293 df-irng 33699 |
This theorem is referenced by: algextdeglem4 33726 |
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