fldextrspundgle - Mathbox for Thierry Arnoux

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Theorem fldextrspundgle 33862

Description: Inequality involving the degree of two different field extensions 𝐼 and 𝐽 of a same field 𝐹. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)

**Hypotheses**
Ref	Expression
fldextrspunfld.k	⊢ 𝐾 = (𝐿 ↾_s 𝐹)
fldextrspunfld.i	⊢ 𝐼 = (𝐿 ↾_s 𝐺)
fldextrspunfld.j	⊢ 𝐽 = (𝐿 ↾_s 𝐻)
fldextrspunfld.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspunfld.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspunfld.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspunfld.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspunfld.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldextrspunfld.7	⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
fldextrspundgle.1	⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻)))

**Assertion**
Ref	Expression
fldextrspundgle	⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))

**Proof of Theorem fldextrspundgle**
Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
2		fldextrspunfld.i	. . . 4 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
3		fldextrspundgle.1	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻)))
4		fldextrspunfld.2	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
5		fldextrspunfld.5	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
7	1	sdrgss 20765	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
9	1, 2, 3, 4, 5, 8	fldgenfldext 33852	. . 3 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgval 33837	. . 3 ⊢ (𝐸/_FldExt𝐼 → (𝐸[:]𝐼) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))))
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (𝐸[:]𝐼) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))))
12		fldextrspunfld.k	. . . . . . . . 9 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
13		fldextrspunfld.j	. . . . . . . . 9 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
14		fldextrspunfld.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
15		fldextrspunfld.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
16		fldextrspunfld.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
17		eqid 2739	. . . . . . . . 9 ⊢ (RingSpan‘𝐿) = (RingSpan‘𝐿)
18		eqid 2739	. . . . . . . . 9 ⊢ ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))
19		eqid 2739	. . . . . . . . 9 ⊢ (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)))
20	12, 2, 13, 4, 14, 15, 5, 6, 16, 17, 18, 19	fldextrspunlem2 33861	. . . . . . . 8 ⊢ (𝜑 → ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	20	oveq2d 7372	. . . . . . 7 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))))
22	21, 3	eqtr4di 2792	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = 𝐸)
23	22	fveq2d 6831	. . . . 5 ⊢ (𝜑 → (subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)))) = (subringAlg ‘𝐸))
24	1	sdrgss 20765	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
25	2, 1	ressbas2 17199	. . . . . 6 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
26	5, 24, 25	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
27	23, 26	fveq12d 6834	. . . 4 ⊢ (𝜑 → ((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺) = ((subringAlg ‘𝐸)‘(Base‘𝐼)))
28	27	fveq2d 6831	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺)) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))))
29	12, 2, 13, 4, 14, 15, 5, 6, 16, 17, 18, 19	fldextrspunlem1 33859	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺)) ≤ (𝐽[:]𝐾))
30	28, 29	eqbrtrrd 5096	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))) ≤ (𝐽[:]𝐾))
31	11, 30	eqbrtrd 5094	1 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ∪ cun 3881 ⊆ wss 3883 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 ≤ cle 11171 ℕ₀cn0 12428 Basecbs 17170 ↾_s cress 17191 RingSpancrgspn 20582 Fieldcfield 20702 SubDRingcsdrg 20758 subringAlg csra 21161 fldGen cfldgen 33394 dimcldim 33783 /_FldExtcfldext 33822 [:]cextdg 33824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-reg 9497 ax-inf2 9553 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-rpss 7666 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-inf 9346 df-oi 9415 df-r1 9679 df-rank 9680 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-ind 12151 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-xadd 13055 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-word 14467 df-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-s2 14801 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ocomp 17232 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-mri 17541 df-acs 17542 df-proset 18251 df-drs 18252 df-poset 18270 df-ipo 18485 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cntr 19284 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20485 df-subrng 20518 df-subrg 20542 df-rgspn 20583 df-rlreg 20666 df-domn 20667 df-idom 20668 df-drng 20703 df-field 20704 df-sdrg 20759 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lmhm 21012 df-lmim 21013 df-lbs 21065 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-cnfld 21348 df-zring 21422 df-dsmm 21707 df-frlm 21722 df-uvc 21758 df-lindf 21781 df-linds 21782 df-assa 21828 df-fldgen 33395 df-dim 33784 df-fldext 33825 df-extdg 33826

This theorem is referenced by: fldextrspundglemul 33863

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