fldextrspundglemul - Mathbox for Thierry Arnoux

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Theorem fldextrspundglemul 33829

Description: Given two field extensions 𝐼 / 𝐾 and 𝐽 / 𝐾 of the same field 𝐾, 𝐽 / 𝐾 being finite, and the composiste field 𝐸 = 𝐼𝐽, the degree of the extension of the composite field 𝐸 / 𝐾 is at most the product of the field extension degrees of 𝐼 / 𝐾 and 𝐽 / 𝐾. (Contributed by Thierry Arnoux, 19-Oct-2025.)

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

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

**Proof of Theorem fldextrspundglemul**
Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
2		fldextrspun.i	. . . . 5 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
3		fldextrspundglemul.1	. . . . 5 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻)))
4		fldextrspun.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
5		fldextrspun.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		fldextrspun.6	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
7	1	sdrgss 20728	. . . . . 6 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
9	1, 2, 3, 4, 5, 8	fldgenfldext 33818	. . . 4 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgcl 33806	. . . 4 ⊢ (𝐸/_FldExt𝐼 → (𝐸[:]𝐼) ∈ ℕ₀^*)
11		xnn0xr 12480	. . . 4 ⊢ ((𝐸[:]𝐼) ∈ ℕ₀^* → (𝐸[:]𝐼) ∈ ℝ^*)
12	9, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ^*)
13		fldextrspun.j	. . . . 5 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
14		fldextrspun.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
15		fldextrspun.k	. . . . 5 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
16	13, 4, 6, 14, 15	fldsdrgfldext2 33812	. . . 4 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
17		extdgcl 33806	. . . 4 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) ∈ ℕ₀^*)
18		xnn0xr 12480	. . . 4 ⊢ ((𝐽[:]𝐾) ∈ ℕ₀^* → (𝐽[:]𝐾) ∈ ℝ^*)
19	16, 17, 18	3syl 18	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ^*)
20		fldextrspun.3	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
21	2, 4, 5, 20, 15	fldsdrgfldext2 33812	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
22		extdgcl 33806	. . . . 5 ⊢ (𝐼/_FldExt𝐾 → (𝐼[:]𝐾) ∈ ℕ₀^*)
23		xnn0xrge0 13423	. . . . 5 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ (0[,]+∞))
24	21, 22, 23	3syl 18	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ (0[,]+∞))
25		elxrge0 13374	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ (0[,]+∞) ↔ ((𝐼[:]𝐾) ∈ ℝ^* ∧ 0 ≤ (𝐼[:]𝐾)))
26	24, 25	sylib 218	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ∈ ℝ^* ∧ 0 ≤ (𝐼[:]𝐾)))
27		fldextrspundglemul.7	. . . 4 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
28	15, 2, 13, 4, 20, 14, 5, 6, 27, 3	fldextrspundgle 33828	. . 3 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))
29		xlemul1a 13204	. . 3 ⊢ ((((𝐸[:]𝐼) ∈ ℝ^* ∧ (𝐽[:]𝐾) ∈ ℝ^* ∧ ((𝐼[:]𝐾) ∈ ℝ^* ∧ 0 ≤ (𝐼[:]𝐾))) ∧ (𝐸[:]𝐼) ≤ (𝐽[:]𝐾)) → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) ≤ ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
30	12, 19, 26, 28, 29	syl31anc 1376	. 2 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) ≤ ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
31		extdgmul 33813	. . 3 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
32	9, 21, 31	syl2anc 585	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
33		xnn0xr 12480	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ ℝ^*)
34	21, 22, 33	3syl 18	. . 3 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ^*)
35		xmulcom 13182	. . 3 ⊢ (((𝐼[:]𝐾) ∈ ℝ^* ∧ (𝐽[:]𝐾) ∈ ℝ^*) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
36	34, 19, 35	syl2anc 585	. 2 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
37	30, 32, 36	3brtr4d 5118	1 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∪ cun 3888 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6490 (class class class)co 7358 0cc0 11027 +∞cpnf 11164 ℝ^*cxr 11166 ≤ cle 11168 ℕ₀cn0 12402 ℕ₀^*cxnn0 12475 ·_e cxmu 13026 [,]cicc 13265 Basecbs 17137 ↾_s cress 17158 Fieldcfield 20665 SubDRingcsdrg 20721 fldGen cfldgen 33376 /_FldExtcfldext 33788 [:]cextdg 33790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-reg 9498 ax-inf2 9551 ax-ac2 10374 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 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 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-inf 9347 df-oi 9416 df-r1 9677 df-rank 9678 df-dju 9814 df-card 9852 df-acn 9855 df-ac 10027 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12609 df-uz 12753 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-icc 13269 df-fz 13425 df-fzo 13572 df-seq 13926 df-exp 13986 df-hash 14255 df-word 14438 df-lsw 14487 df-concat 14495 df-s1 14521 df-substr 14566 df-pfx 14596 df-s2 14772 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15412 df-sum 15611 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ocomp 17199 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-0g 17362 df-gsum 17363 df-prds 17368 df-pws 17370 df-mre 17506 df-mrc 17507 df-mri 17508 df-acs 17509 df-proset 18218 df-drs 18219 df-poset 18237 df-ipo 18452 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-cntr 19251 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-nzr 20448 df-subrng 20481 df-subrg 20505 df-rgspn 20546 df-rlreg 20629 df-domn 20630 df-idom 20631 df-drng 20666 df-field 20667 df-sdrg 20722 df-lmod 20815 df-lss 20885 df-lsp 20925 df-lmhm 20976 df-lmim 20977 df-lbs 21029 df-lvec 21057 df-sra 21127 df-rgmod 21128 df-cnfld 21312 df-zring 21404 df-dsmm 21689 df-frlm 21704 df-uvc 21740 df-lindf 21763 df-linds 21764 df-assa 21810 df-ind 32913 df-fldgen 33377 df-dim 33749 df-fldext 33791 df-extdg 33792

This theorem is referenced by: fldextrspundgdvdslem 33830 fldextrspundgdvds 33831 fldext2rspun 33832

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