finexttrb - Mathbox for Thierry Arnoux

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Theorem finexttrb 33858

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 33856	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)))
2	1	eleq1d 2824	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸[:]𝐾) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
3		fldexttr 33851	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 𝐸/_FldExt𝐾)
4		brfinext 33845	. . 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 33845	. . . 4 ⊢ (𝐸/_FldExt𝐹 → (𝐸/_FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ₀))
7		brfinext 33845	. . . 4 ⊢ (𝐹/_FldExt𝐾 → (𝐹/_FinExt𝐾 ↔ (𝐹[:]𝐾) ∈ ℕ₀))
8	6, 7	bi2anan9 644	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾) ↔ ((𝐸[:]𝐹) ∈ ℕ₀ ∧ (𝐹[:]𝐾) ∈ ℕ₀)))
9		extdgcl 33849	. . . . 5 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ₀^*)
10	9	adantr 481	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ∈ ℕ₀^*)
11		extdgcl 33849	. . . . 5 ⊢ (𝐹/_FldExt𝐾 → (𝐹[:]𝐾) ∈ ℕ₀^*)
12	11	adantl 482	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ∈ ℕ₀^*)
13		extdggt0 33850	. . . . . 6 ⊢ (𝐸/_FldExt𝐹 → 0 < (𝐸[:]𝐹))
14	13	adantr 481	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐸[:]𝐹))
15	14	gt0ne0d 11706	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸[:]𝐹) ≠ 0)
16		extdggt0 33850	. . . . . 6 ⊢ (𝐹/_FldExt𝐾 → 0 < (𝐹[:]𝐾))
17	16	adantl 482	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → 0 < (𝐹[:]𝐾))
18	17	gt0ne0d 11706	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐹[:]𝐾) ≠ 0)
19		nn0xmulclb 32864	. . . 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 844	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀ ↔ ((𝐸[:]𝐹) ∈ ℕ₀ ∧ (𝐹[:]𝐾) ∈ ℕ₀)))
21	8, 20	bitr4d 283	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → ((𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾) ↔ ((𝐸[:]𝐹) ·_e (𝐹[:]𝐾)) ∈ ℕ₀))
22	2, 5, 21	3bitr4d 312	1 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝐹/_FldExt𝐾) → (𝐸/_FinExt𝐾 ↔ (𝐸/_FinExt𝐹 ∧ 𝐹/_FinExt𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2119 ≠ wne 2934 class class class wbr 5073 (class class class)co 7357 0cc0 11030 < clt 11171 ℕ₀cn0 12429 ℕ₀^*cxnn0 12502 ·_e cxmu 13054 /_FldExtcfldext 33831 /_FinExtcfinext 33832 [:]cextdg 33833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-reg 9498 ax-inf2 9554 ax-ac2 10377 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-rpss 7667 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-r1 9680 df-rank 9681 df-dju 9817 df-card 9855 df-acn 9858 df-ac 10030 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-xnn0 12503 df-z 12517 df-dec 12637 df-uz 12781 df-xmul 13057 df-fz 13454 df-fzo 13601 df-seq 13956 df-hash 14285 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ocomp 17233 df-ds 17234 df-hom 17236 df-cco 17237 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17540 df-mrc 17541 df-mri 17542 df-acs 17543 df-proset 18252 df-drs 18253 df-poset 18271 df-ipo 18486 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18743 df-submnd 18744 df-grp 18904 df-minusg 18905 df-sbg 18906 df-mulg 19036 df-subg 19091 df-ghm 19180 df-cntz 19284 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-oppr 20309 df-dvdsr 20329 df-unit 20330 df-invr 20360 df-nzr 20486 df-subrng 20519 df-subrg 20543 df-drng 20704 df-field 20705 df-lmod 20853 df-lss 20923 df-lsp 20963 df-lmhm 21013 df-lbs 21066 df-lvec 21094 df-sra 21164 df-rgmod 21165 df-dsmm 21708 df-frlm 21723 df-uvc 21759 df-lindf 21782 df-linds 21783 df-dim 33793 df-fldext 33834 df-extdg 33835 df-finext 33836

This theorem is referenced by: (None)

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