finexttrb - Mathbox for Thierry Arnoux

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Theorem finexttrb 33667

Description: The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.)

**Assertion**
Ref	Expression
finexttrb	⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸/_FinExt𝐾 ↔ (𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾)))

**Proof of Theorem finexttrb**
Step	Hyp	Ref	Expression
1		extdgmul 33666	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)))
2	1	eleq1d 2814	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸[:]𝐾) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
3		fldexttr 33661	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 𝐸/_FldExt𝐾)
4		brfinext 33655	. . 3 ⊢ (𝐸/_FldExt𝐾 → (𝐸/_FinExt𝐾 ↔ (𝐸[:]𝐾) ∈ ℕ₀))
5	3, 4	syl 17	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸/_FinExt𝐾 ↔ (𝐸[:]𝐾) ∈ ℕ₀))
6		brfinext 33655	. . . 4 ⊢ (𝐸/_FldExt𝐹 → (𝐸/_FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ₀))
7		brfinext 33655	. . . 4 ⊢ (𝐹/_FldExt𝐾 → (𝐹/_FinExt𝐾 ↔ (𝐹[:]𝐾) ∈ ℕ₀))
8	6, 7	bi2anan9 638	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾) ↔ ((𝐸[:]𝐹) ∈ ℕ₀ ∧ (𝐹[:]𝐾) ∈ ℕ₀)))
9		extdgcl 33659	. . . . 5 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ₀^*)
10	9	adantr 480	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ∈ ℕ₀^*)
11		extdgcl 33659	. . . . 5 ⊢ (𝐹/_FldExt𝐾 → (𝐹[:]𝐾) ∈ ℕ₀^*)
12	11	adantl 481	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ∈ ℕ₀^*)
13		extdggt0 33660	. . . . . 6 ⊢ (𝐸/_FldExt𝐹 → 0 < (𝐸[:]𝐹))
14	13	adantr 480	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐸[:]𝐹))
15	14	gt0ne0d 11749	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ≠ 0)
16		extdggt0 33660	. . . . . 6 ⊢ (𝐹/_FldExt𝐾 → 0 < (𝐹[:]𝐾))
17	16	adantl 481	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐹[:]𝐾))
18	17	gt0ne0d 11749	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ≠ 0)
19		nn0xmulclb 32701	. . . 4 ⊢ ((((𝐸[:]𝐹) ∈ ℕ₀^* ∧ (𝐹[:]𝐾) ∈ ℕ₀^*) ∧ ((𝐸[:]𝐹) ≠ 0 ∧ (𝐹[:]𝐾) ≠ 0)) → (((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ∈ ℕ₀ ∧ (𝐹[:]𝐾) ∈ ℕ₀)))
20	10, 12, 15, 18, 19	syl22anc 838	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ∈ ℕ₀ ∧ (𝐹[:]𝐾) ∈ ℕ₀)))
21	8, 20	bitr4d 282	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾) ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
22	2, 5, 21	3bitr4d 311	1 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸/_FinExt𝐾 ↔ (𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5110 (class class class)co 7390 0cc0 11075 < clt 11215 ℕ₀cn0 12449 ℕ₀^*cxnn0 12522 ·_e cxmu 13078 /_FldExtcfldext 33641 /_FinExtcfinext 33642 [:]cextdg 33643

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-reg 9552 ax-inf2 9601 ax-ac2 10423 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-rpss 7702 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-r1 9724 df-rank 9725 df-dju 9861 df-card 9899 df-acn 9902 df-ac 10076 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-xmul 13081 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ocomp 17248 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mre 17554 df-mrc 17555 df-mri 17556 df-acs 17557 df-proset 18262 df-drs 18263 df-poset 18281 df-ipo 18494 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-nzr 20429 df-subrng 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lmhm 20936 df-lbs 20989 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-dsmm 21648 df-frlm 21663 df-uvc 21699 df-lindf 21722 df-linds 21723 df-dim 33602 df-fldext 33644 df-extdg 33645 df-finext 33646

This theorem is referenced by: (None)

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