fldextrspundgle - Mathbox for Thierry Arnoux

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Theorem fldextrspundgle 33651

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 20678	. . . . 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 33641	. . 3 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgval 33626	. . 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 33650	. . . . . . . 8 ⊢ (𝜑 → ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	20	oveq2d 7365	. . . . . . 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 6826	. . . . 5 ⊢ (𝜑 → (subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻)))) = (subringAlg ‘𝐸))
24	1	sdrgss 20678	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
25	2, 1	ressbas2 17149	. . . . . 6 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
26	5, 24, 25	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
27	23, 26	fveq12d 6829	. . . 4 ⊢ (𝜑 → ((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺) = ((subringAlg ‘𝐸)‘(Base‘𝐼)))
28	27	fveq2d 6826	. . 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 33648	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘(𝐿 ↾_s ((RingSpan‘𝐿)‘(𝐺 ∪ 𝐻))))‘𝐺)) ≤ (𝐽[:]𝐾))
30	28, 29	eqbrtrrd 5116	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐼))) ≤ (𝐽[:]𝐾))
31	11, 30	eqbrtrd 5114	1 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∪ cun 3901 ⊆ wss 3903 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ≤ cle 11150 ℕ₀cn0 12384 Basecbs 17120 ↾_s cress 17141 RingSpancrgspn 20495 Fieldcfield 20615 SubDRingcsdrg 20671 subringAlg csra 21075 fldGen cfldgen 33250 dimcldim 33571 /_FldExtcfldext 33611 [:]cextdg 33613

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-reg 9484 ax-inf2 9537 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-rpss 7659 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-inf 9333 df-oi 9402 df-r1 9660 df-rank 9661 df-dju 9797 df-card 9835 df-acn 9838 df-ac 10010 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-xadd 13015 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14503 df-substr 14548 df-pfx 14578 df-s2 14755 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ocomp 17182 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-mri 17490 df-acs 17491 df-proset 18200 df-drs 18201 df-poset 18219 df-ipo 18434 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-mulg 18947 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cntr 19197 df-lsm 19515 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-nzr 20398 df-subrng 20431 df-subrg 20455 df-rgspn 20496 df-rlreg 20579 df-domn 20580 df-idom 20581 df-drng 20616 df-field 20617 df-sdrg 20672 df-lmod 20765 df-lss 20835 df-lsp 20875 df-lmhm 20926 df-lmim 20927 df-lbs 20979 df-lvec 21007 df-sra 21077 df-rgmod 21078 df-cnfld 21262 df-zring 21354 df-dsmm 21639 df-frlm 21654 df-uvc 21690 df-lindf 21713 df-linds 21714 df-assa 21760 df-ind 32795 df-fldgen 33251 df-dim 33572 df-fldext 33614 df-extdg 33615

This theorem is referenced by: fldextrspundglemul 33652

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