fldextrspundglemul - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextrspundglemul		Structured version Visualization version GIF version

Theorem fldextrspundglemul 33856

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 20738	. . . . . 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 33845	. . . 4 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		extdgcl 33833	. . . 4 ⊢ (𝐸/_FldExt𝐼 → (𝐸[:]𝐼) ∈ ℕ₀^*)
11		xnn0xr 12491	. . . 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 33839	. . . 4 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
17		extdgcl 33833	. . . 4 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) ∈ ℕ₀^*)
18		xnn0xr 12491	. . . 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 33839	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
22		extdgcl 33833	. . . . 5 ⊢ (𝐼/_FldExt𝐾 → (𝐼[:]𝐾) ∈ ℕ₀^*)
23		xnn0xrge0 13434	. . . . 5 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ (0[,]+∞))
24	21, 22, 23	3syl 18	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ (0[,]+∞))
25		elxrge0 13385	. . . 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 33855	. . 3 ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))
29		xlemul1a 13215	. . 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 33840	. . 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 12491	. . . 4 ⊢ ((𝐼[:]𝐾) ∈ ℕ₀^* → (𝐼[:]𝐾) ∈ ℝ^*)
34	21, 22, 33	3syl 18	. . 3 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ^*)
35		xmulcom 13193	. . 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 5132	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 3901 ⊆ wss 3903 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 0cc0 11038 +∞cpnf 11175 ℝ^*cxr 11177 ≤ cle 11179 ℕ₀cn0 12413 ℕ₀^*cxnn0 12486 ·_e cxmu 13037 [,]cicc 13276 Basecbs 17148 ↾_s cress 17169 Fieldcfield 20675 SubDRingcsdrg 20731 fldGen cfldgen 33403 /_FldExtcfldext 33815 [:]cextdg 33817

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-word 14449 df-lsw 14498 df-concat 14506 df-s1 14532 df-substr 14577 df-pfx 14607 df-s2 14783 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cntr 19259 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rgspn 20556 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-sdrg 20732 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lmim 20987 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-cnfld 21322 df-zring 21414 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-assa 21820 df-ind 32940 df-fldgen 33404 df-dim 33776 df-fldext 33818 df-extdg 33819

This theorem is referenced by: fldextrspundgdvdslem 33857 fldextrspundgdvds 33858 fldext2rspun 33859

W3C validator