fldextrspundgle - Mathbox for Thierry Arnoux

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Theorem fldextrspundgle 33835

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 2736	. . . 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 20726	. . . . 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 33825	. . 3 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgval 33810	. . 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 2736	. . . . . . . . 9 ⊢ (RingSpan‘𝐿) = (RingSpan‘𝐿)
18		eqid 2736	. . . . . . . . 9 ⊢ ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))
19		eqid 2736	. . . . . . . . 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 33834	. . . . . . . 8 ⊢ (𝜑 → ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	20	oveq2d 7374	. . . . . . 7 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))))
22	21, 3	eqtr4di 2789	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = 𝐸)
23	22	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)))) = (subringAlg ‘𝐸))
24	1	sdrgss 20726	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
25	2, 1	ressbas2 17165	. . . . . 6 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
26	5, 24, 25	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
27	23, 26	fveq12d 6841	. . . 4 ⊢ (𝜑 → ((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺) = ((subringAlg ‘𝐸)‘(Base‘𝐼)))
28	27	fveq2d 6838	. . 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 33832	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺)) ≤ (𝐽[:]𝐾))
30	28, 29	eqbrtrrd 5122	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))) ≤ (𝐽[:]𝐾))
31	11, 30	eqbrtrd 5120	1 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ∪ cun 3899 ⊆ wss 3901 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ≤ cle 11167 ℕ₀cn0 12401 Basecbs 17136 ↾_s cress 17157 RingSpancrgspn 20543 Fieldcfield 20663 SubDRingcsdrg 20719 subringAlg csra 21123 fldGen cfldgen 33392 dimcldim 33755 /_FldExtcfldext 33795 [:]cextdg 33797

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-reg 9497 ax-inf2 9550 ax-ac2 10373 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 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 9676 df-rank 9677 df-dju 9813 df-card 9851 df-acn 9854 df-ac 10026 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-xadd 13027 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-word 14437 df-lsw 14486 df-concat 14494 df-s1 14520 df-substr 14565 df-pfx 14595 df-s2 14771 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ocomp 17198 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-mri 17507 df-acs 17508 df-proset 18217 df-drs 18218 df-poset 18236 df-ipo 18451 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cntr 19247 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-subrng 20479 df-subrg 20503 df-rgspn 20544 df-rlreg 20627 df-domn 20628 df-idom 20629 df-drng 20664 df-field 20665 df-sdrg 20720 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lmhm 20974 df-lmim 20975 df-lbs 21027 df-lvec 21055 df-sra 21125 df-rgmod 21126 df-cnfld 21310 df-zring 21402 df-dsmm 21687 df-frlm 21702 df-uvc 21738 df-lindf 21761 df-linds 21762 df-assa 21808 df-ind 32930 df-fldgen 33393 df-dim 33756 df-fldext 33798 df-extdg 33799

This theorem is referenced by: fldextrspundglemul 33836

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