fldextrspundglemul - Mathbox for Thierry Arnoux

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Theorem fldextrspundglemul 33870

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 2740	. . . . 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 20772	. . . . . 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 33859	. . . 4 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgcl 33847	. . . 4 ⊢ (𝐸/_FldExt𝐼 → (𝐸[:]𝐼) ∈ ℕ₀^*)
11		xnn0xr 12513	. . . 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 33853	. . . 4 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
17		extdgcl 33847	. . . 4 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) ∈ ℕ₀^*)
18		xnn0xr 12513	. . . 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 33853	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
22		extdgcl 33847	. . . . 5 ⊢ (𝐼/_FldExt𝐾 → (𝐼[:]𝐾) ∈ ℕ₀^*)
23		xnn0xrge0 13457	. . . . 5 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ (0[,]+∞))
24	21, 22, 23	3syl 18	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ (0[,]+∞))
25		elxrge0 13408	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ (0[,]+∞) ↔ ((𝐼[:]𝐾) ∈ ℝ^* ∧ 0 ≤ (𝐼[:]𝐾)))
26	24, 25	sylib 219	. . 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 33869	. . 3 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))
29		xlemul1a 13238	. . 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 1381	. 2 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) ≤ ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
31		extdgmul 33854	. . 3 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
32	9, 21, 31	syl2anc 590	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
33		xnn0xr 12513	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ ℝ^*)
34	21, 22, 33	3syl 18	. . 3 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ^*)
35		xmulcom 13216	. . 3 ⊢ (((𝐼[:]𝐾) ∈ ℝ^* ∧ (𝐽[:]𝐾) ∈ ℝ^*) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
36	34, 19, 35	syl2anc 590	. 2 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
37	30, 32, 36	3brtr4d 5111	1 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∪ cun 3888 ⊆ wss 3890 class class class wbr 5079 ‘cfv 6492 (class class class)co 7363 0cc0 11036 +∞cpnf 11174 ℝ^*cxr 11176 ≤ cle 11178 ℕ₀cn0 12435 ℕ₀^*cxnn0 12508 ·_e cxmu 13060 [,]cicc 13299 Basecbs 17177 ↾_s cress 17198 Fieldcfield 20709 SubDRingcsdrg 20765 fldGen cfldgen 33401 /_FldExtcfldext 33829 [:]cextdg 33831

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 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-reg 9504 ax-inf2 9560 ax-ac2 10383 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

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 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-rpss 7673 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-tpos 8173 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-inf 9353 df-oi 9422 df-r1 9686 df-rank 9687 df-dju 9823 df-card 9861 df-acn 9864 df-ac 10036 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-ind 12158 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-xnn0 12509 df-z 12523 df-dec 12643 df-uz 12787 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-icc 13303 df-fz 13460 df-fzo 13607 df-seq 13962 df-exp 14022 df-hash 14291 df-word 14474 df-lsw 14523 df-concat 14531 df-s1 14557 df-substr 14602 df-pfx 14632 df-s2 14808 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-clim 15448 df-sum 15647 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ocomp 17239 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-0g 17402 df-gsum 17403 df-prds 17408 df-pws 17410 df-mre 17546 df-mrc 17547 df-mri 17548 df-acs 17549 df-proset 18258 df-drs 18259 df-poset 18277 df-ipo 18492 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-ghm 19186 df-cntz 19290 df-cntr 19291 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20492 df-subrng 20525 df-subrg 20549 df-rgspn 20590 df-rlreg 20673 df-domn 20674 df-idom 20675 df-drng 20710 df-field 20711 df-sdrg 20766 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lmhm 21019 df-lmim 21020 df-lbs 21072 df-lvec 21100 df-sra 21170 df-rgmod 21171 df-cnfld 21355 df-zring 21429 df-dsmm 21714 df-frlm 21729 df-uvc 21765 df-lindf 21788 df-linds 21789 df-assa 21835 df-fldgen 33402 df-dim 33791 df-fldext 33832 df-extdg 33833

This theorem is referenced by: fldextrspundgdvdslem 33871 fldextrspundgdvds 33872 fldext2rspun 33873

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