fldextrspundglemul - Mathbox for Thierry Arnoux

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Theorem fldextrspundglemul 33930

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 2756	. . . . 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 20815	. . . . . 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 33919	. . . 4 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgcl 33907	. . . 4 ⊢ (𝐸/_FldExt𝐼 → (𝐸[:]𝐼) ∈ ℕ₀^*)
11		xnn0xr 12549	. . . 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 33913	. . . 4 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
17		extdgcl 33907	. . . 4 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) ∈ ℕ₀^*)
18		xnn0xr 12549	. . . 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 33913	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
22		extdgcl 33907	. . . . 5 ⊢ (𝐼/_FldExt𝐾 → (𝐼[:]𝐾) ∈ ℕ₀^*)
23		xnn0xrge0 13500	. . . . 5 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ (0[,]+∞))
24	21, 22, 23	3syl 18	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ (0[,]+∞))
25		elxrge0 13451	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ (0[,]+∞) ↔ ((𝐼[:]𝐾) ∈ ℝ^* ∧ 0 ≤ (𝐼[:]𝐾)))
26	24, 25	sylib 220	. . 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 33929	. . 3 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))
29		xlemul1a 13281	. . 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 1388	. 2 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) ≤ ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
31		extdgmul 33914	. . 3 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
32	9, 21, 31	syl2anc 592	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
33		xnn0xr 12549	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ ℝ^*)
34	21, 22, 33	3syl 18	. . 3 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ^*)
35		xmulcom 13259	. . 3 ⊢ (((𝐼[:]𝐾) ∈ ℝ^* ∧ (𝐽[:]𝐾) ∈ ℝ^*) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
36	34, 19, 35	syl2anc 592	. 2 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐽[:]𝐾) ·_e (𝐼[:]𝐾)))
37	30, 32, 36	3brtr4d 5126	1 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1554 ∈ wcel 2136 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5094 ‘cfv 6510 (class class class)co 7385 0cc0 11063 +∞cpnf 11203 ℝ^*cxr 11205 ≤ cle 11207 ℕ₀cn0 12471 ℕ₀^*cxnn0 12544 ·_e cxmu 13103 [,]cicc 13342 Basecbs 17221 ↾_s cress 17242 Fieldcfield 20752 SubDRingcsdrg 20808 fldGen cfldgen 33451 /_FldExtcfldext 33889 [:]cextdg 33891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-reg 9530 ax-inf2 9586 ax-ac2 10410 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-rpss 7695 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-tpos 8194 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-oadd 8429 df-er 8666 df-map 8798 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-sup 9378 df-inf 9379 df-oi 9448 df-r1 9712 df-rank 9713 df-dju 9849 df-card 9887 df-acn 9890 df-ac 10062 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-ind 12186 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-xnn0 12545 df-z 12559 df-dec 12679 df-uz 12830 df-rp 12984 df-xneg 13104 df-xadd 13105 df-xmul 13106 df-icc 13346 df-fz 13503 df-fzo 13650 df-seq 14005 df-exp 14065 df-hash 14334 df-word 14517 df-lsw 14566 df-concat 14574 df-s1 14600 df-substr 14645 df-pfx 14675 df-s2 14851 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-clim 15491 df-sum 15690 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ocomp 17283 df-ds 17284 df-unif 17285 df-hom 17286 df-cco 17287 df-0g 17446 df-gsum 17447 df-prds 17452 df-pws 17454 df-mre 17590 df-mrc 17591 df-mri 17592 df-acs 17593 df-proset 18302 df-drs 18303 df-poset 18321 df-ipo 18536 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19230 df-cntz 19333 df-cntr 19334 df-lsm 19652 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-nzr 20535 df-subrng 20568 df-subrg 20592 df-rgspn 20633 df-rlreg 20716 df-domn 20717 df-idom 20718 df-drng 20753 df-field 20754 df-sdrg 20809 df-lmod 20902 df-lss 20972 df-lsp 21012 df-lmhm 21062 df-lmim 21063 df-lbs 21115 df-lvec 21143 df-sra 21213 df-rgmod 21214 df-cnfld 21398 df-zring 21472 df-dsmm 21757 df-frlm 21772 df-uvc 21808 df-lindf 21831 df-linds 21832 df-assa 21878 df-fldgen 33452 df-dim 33851 df-fldext 33892 df-extdg 33893

This theorem is referenced by: fldextrspundgdvdslem 33931 fldextrspundgdvds 33932 fldext2rspun 33933

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