fldextrspundgle - Mathbox for Thierry Arnoux

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Theorem fldextrspundgle 33664

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 2729	. . . 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 20697	. . . . 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 33654	. . 3 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgval 33639	. . 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 2729	. . . . . . . . 9 ⊢ (RingSpan‘𝐿) = (RingSpan‘𝐿)
18		eqid 2729	. . . . . . . . 9 ⊢ ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))
19		eqid 2729	. . . . . . . . 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 33663	. . . . . . . 8 ⊢ (𝜑 → ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	20	oveq2d 7369	. . . . . . 7 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))))
22	21, 3	eqtr4di 2782	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))) = 𝐸)
23	22	fveq2d 6830	. . . . 5 ⊢ (𝜑 → (subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)))) = (subringAlg ‘𝐸))
24	1	sdrgss 20697	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
25	2, 1	ressbas2 17168	. . . . . 6 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
26	5, 24, 25	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
27	23, 26	fveq12d 6833	. . . 4 ⊢ (𝜑 → ((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺) = ((subringAlg ‘𝐸)‘(Base‘𝐼)))
28	27	fveq2d 6830	. . 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 33661	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺)) ≤ (𝐽[:]𝐾))
30	28, 29	eqbrtrrd 5119	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))) ≤ (𝐽[:]𝐾))
31	11, 30	eqbrtrd 5117	1 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∪ cun 3903 ⊆ wss 3905 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ≤ cle 11169 ℕ₀cn0 12403 Basecbs 17139 ↾_s cress 17160 RingSpancrgspn 20514 Fieldcfield 20634 SubDRingcsdrg 20690 subringAlg csra 21094 fldGen cfldgen 33268 dimcldim 33584 /_FldExtcfldext 33624 [:]cextdg 33626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-reg 9503 ax-inf2 9556 ax-ac2 10376 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-rpss 7663 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 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-inf 9352 df-oi 9421 df-r1 9679 df-rank 9680 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12611 df-uz 12755 df-rp 12913 df-xadd 13034 df-fz 13430 df-fzo 13577 df-seq 13928 df-exp 13988 df-hash 14257 df-word 14440 df-lsw 14489 df-concat 14497 df-s1 14522 df-substr 14567 df-pfx 14597 df-s2 14774 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-clim 15414 df-sum 15613 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ocomp 17201 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-0g 17364 df-gsum 17365 df-prds 17370 df-pws 17372 df-mre 17507 df-mrc 17508 df-mri 17509 df-acs 17510 df-proset 18219 df-drs 18220 df-poset 18238 df-ipo 18453 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-mhm 18676 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-mulg 18966 df-subg 19021 df-ghm 19111 df-cntz 19215 df-cntr 19216 df-lsm 19534 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-cring 20140 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-nzr 20417 df-subrng 20450 df-subrg 20474 df-rgspn 20515 df-rlreg 20598 df-domn 20599 df-idom 20600 df-drng 20635 df-field 20636 df-sdrg 20691 df-lmod 20784 df-lss 20854 df-lsp 20894 df-lmhm 20945 df-lmim 20946 df-lbs 20998 df-lvec 21026 df-sra 21096 df-rgmod 21097 df-cnfld 21281 df-zring 21373 df-dsmm 21658 df-frlm 21673 df-uvc 21709 df-lindf 21732 df-linds 21733 df-assa 21779 df-ind 32813 df-fldgen 33269 df-dim 33585 df-fldext 33627 df-extdg 33628

This theorem is referenced by: fldextrspundglemul 33665

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