finexttrb - Mathbox for Thierry Arnoux

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Theorem finexttrb 33678

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 33676	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)))
2	1	eleq1d 2816	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸[:]𝐾) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
3		fldexttr 33671	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 𝐸/_FldExt𝐾)
4		brfinext 33665	. . 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 33665	. . . 4 ⊢ (𝐸/_FldExt𝐹 → (𝐸/_FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ₀))
7		brfinext 33665	. . . 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 33669	. . . . 5 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ₀^*)
10	9	adantr 480	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ∈ ℕ₀^*)
11		extdgcl 33669	. . . . 5 ⊢ (𝐹/_FldExt𝐾 → (𝐹[:]𝐾) ∈ ℕ₀^*)
12	11	adantl 481	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ∈ ℕ₀^*)
13		extdggt0 33670	. . . . . 6 ⊢ (𝐸/_FldExt𝐹 → 0 < (𝐸[:]𝐹))
14	13	adantr 480	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐸[:]𝐹))
15	14	gt0ne0d 11681	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ≠ 0)
16		extdggt0 33670	. . . . . 6 ⊢ (𝐹/_FldExt𝐾 → 0 < (𝐹[:]𝐾))
17	16	adantl 481	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐹[:]𝐾))
18	17	gt0ne0d 11681	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ≠ 0)
19		nn0xmulclb 32754	. . . 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 2111 ≠ wne 2928 class class class wbr 5089 (class class class)co 7346 0cc0 11006 < clt 11146 ℕ₀cn0 12381 ℕ₀^*cxnn0 12454 ·_e cxmu 13010 /_FldExtcfldext 33651 /_FinExtcfinext 33652 [:]cextdg 33653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-reg 9478 ax-inf2 9531 ax-ac2 10354 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-rpss 7656 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-r1 9657 df-rank 9658 df-dju 9794 df-card 9832 df-acn 9835 df-ac 10007 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-xmul 13013 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ocomp 17182 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-mri 17490 df-acs 17491 df-proset 18200 df-drs 18201 df-poset 18219 df-ipo 18434 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-drng 20646 df-field 20647 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lmhm 20956 df-lbs 21009 df-lvec 21037 df-sra 21107 df-rgmod 21108 df-dsmm 21669 df-frlm 21684 df-uvc 21720 df-lindf 21743 df-linds 21744 df-dim 33612 df-fldext 33654 df-extdg 33655 df-finext 33656

This theorem is referenced by: (None)

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