finexttrb - Mathbox for Thierry Arnoux

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Theorem finexttrb 33706

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 33705	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)))
2	1	eleq1d 2819	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸[:]𝐾) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
3		fldexttr 33700	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 𝐸/_FldExt𝐾)
4		brfinext 33694	. . 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 33694	. . . 4 ⊢ (𝐸/_FldExt𝐹 → (𝐸/_FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ₀))
7		brfinext 33694	. . . 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 33698	. . . . 5 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ₀^*)
10	9	adantr 480	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ∈ ℕ₀^*)
11		extdgcl 33698	. . . . 5 ⊢ (𝐹/_FldExt𝐾 → (𝐹[:]𝐾) ∈ ℕ₀^*)
12	11	adantl 481	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ∈ ℕ₀^*)
13		extdggt0 33699	. . . . . 6 ⊢ (𝐸/_FldExt𝐹 → 0 < (𝐸[:]𝐹))
14	13	adantr 480	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐸[:]𝐹))
15	14	gt0ne0d 11801	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ≠ 0)
16		extdggt0 33699	. . . . . 6 ⊢ (𝐹/_FldExt𝐾 → 0 < (𝐹[:]𝐾))
17	16	adantl 481	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐹[:]𝐾))
18	17	gt0ne0d 11801	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ≠ 0)
19		nn0xmulclb 32748	. . . 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 2108 ≠ wne 2932 class class class wbr 5119 (class class class)co 7405 0cc0 11129 < clt 11269 ℕ₀cn0 12501 ℕ₀^*cxnn0 12574 ·_e cxmu 13127 /_FldExtcfldext 33678 /_FinExtcfinext 33679 [:]cextdg 33681

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-reg 9606 ax-inf2 9655 ax-ac2 10477 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-rpss 7717 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-oi 9524 df-r1 9778 df-rank 9779 df-dju 9915 df-card 9953 df-acn 9956 df-ac 10130 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-xmul 13130 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ocomp 17292 df-ds 17293 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-mre 17598 df-mrc 17599 df-mri 17600 df-acs 17601 df-proset 18306 df-drs 18307 df-poset 18325 df-ipo 18538 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-ghm 19196 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-nzr 20473 df-subrng 20506 df-subrg 20530 df-drng 20691 df-field 20692 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lmhm 20980 df-lbs 21033 df-lvec 21061 df-sra 21131 df-rgmod 21132 df-dsmm 21692 df-frlm 21707 df-uvc 21743 df-lindf 21766 df-linds 21767 df-dim 33639 df-fldext 33682 df-extdg 33683 df-finext 33684

This theorem is referenced by: (None)

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