Theorem fldexttr 33764

Description: Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
fldexttr	⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Proof of Theorem fldexttr

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2		simpl 482	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3		fldextfld2 33754	. . . . . . 7 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5		fldextfld2 33754	. . . . . . 7 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
6	1, 5	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7		brfldext 33751	. . . . . 6 ⊢ ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
8	4, 6, 7	syl2anc 584	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
9	1, 8	mpbid 232	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹)))
10	9	simpld 494	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐹 ↾s (Base‘𝐾)))
11		fldextfld1 33753	. . . . . . . . 9 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
12	2, 11	syl 17	. . . . . . . 8 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13		brfldext 33751	. . . . . . . 8 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
14	12, 4, 13	syl2anc 584	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	2, 14	mpbid 232	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
16	15	simpld 494	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
17	16	oveq1d 7371	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)))
18		fvex 6845	. . . . 5 ⊢ (Base‘𝐹) ∈ V
19		fvex 6845	. . . . 5 ⊢ (Base‘𝐾) ∈ V
20		ressress 17172	. . . . 5 ⊢ (((Base‘𝐹) ∈ V ∧ (Base‘𝐾) ∈ V) → ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))))
21	18, 19, 20	mp2an 692	. . . 4 ⊢ ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾)))
22	17, 21	eqtrdi 2785	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))))
23		incom 4159	. . . . 5 ⊢ ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
24	9	simprd 495	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	subrgss 20503	. . . . . . 7 ⊢ ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
27	24, 26	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ⊆ (Base‘𝐹))
28		dfss2 3917	. . . . . 6 ⊢ ((Base‘𝐾) ⊆ (Base‘𝐹) ↔ ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
29	27, 28	sylib 218	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
30	23, 29	eqtr3id 2783	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
31	30	oveq2d 7372	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸 ↾s (Base‘𝐾)))
32	10, 22, 31	3eqtrd 2773	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
33	15	simprd 495	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34	16	fveq2d 6836	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
35	24, 34	eleqtrd 2836	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
36		eqid 2734	. . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
37	36	subsubrg 20529	. . . 4 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
38	37	simprbda 498	. . 3 ⊢ (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
39	33, 35, 38	syl2anc 584	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40		brfldext 33751	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
41	12, 6, 40	syl2anc 584	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
42	32, 39, 41	mpbir2and 713	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134   ↾_s cress 17155  SubRingcsubrg 20500  Fieldcfield 20661  /_FldExtcfldext 33744

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-0g 17359  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-subg 19051  df-mgp 20074  df-ur 20115  df-ring 20168  df-subrg 20501  df-fldext 33747

This theorem is referenced by:  extdgmul  33769  finexttrb  33771  fldext2chn  33834