Theorem fldexttr 31254

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 488	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2		simpl 486	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3		fldextfld2 31246	. . . . . . 7 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5		fldextfld2 31246	. . . . . . 7 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
6	1, 5	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7		brfldext 31243	. . . . . 6 ⊢ ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
8	4, 6, 7	syl2anc 587	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
9	1, 8	mpbid 235	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹)))
10	9	simpld 498	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐹 ↾s (Base‘𝐾)))
11		fldextfld1 31245	. . . . . . . . 9 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
12	2, 11	syl 17	. . . . . . . 8 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13		brfldext 31243	. . . . . . . 8 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
14	12, 4, 13	syl2anc 587	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	2, 14	mpbid 235	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
16	15	simpld 498	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
17	16	oveq1d 7165	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)))
18		fvex 6671	. . . . 5 ⊢ (Base‘𝐹) ∈ V
19		fvex 6671	. . . . 5 ⊢ (Base‘𝐾) ∈ V
20		ressress 16620	. . . . 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 691	. . . 4 ⊢ ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾)))
22	17, 21	eqtrdi 2809	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))))
23		incom 4106	. . . . 5 ⊢ ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
24	9	simprd 499	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25		eqid 2758	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	subrgss 19604	. . . . . . 7 ⊢ ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
27	24, 26	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ⊆ (Base‘𝐹))
28		df-ss 3875	. . . . . 6 ⊢ ((Base‘𝐾) ⊆ (Base‘𝐹) ↔ ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
29	27, 28	sylib 221	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
30	23, 29	syl5eqr 2807	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
31	30	oveq2d 7166	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸 ↾s (Base‘𝐾)))
32	10, 22, 31	3eqtrd 2797	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
33	15	simprd 499	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34	16	fveq2d 6662	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
35	24, 34	eleqtrd 2854	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
36		eqid 2758	. . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
37	36	subsubrg 19629	. . . 4 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
38	37	simprbda 502	. . 3 ⊢ (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
39	33, 35, 38	syl2anc 587	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40		brfldext 31243	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
41	12, 6, 40	syl2anc 587	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
42	32, 39, 41	mpbir2and 712	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150  Basecbs 16541   ↾_s cress 16542  Fieldcfield 19571  SubRingcsubrg 19599  /_FldExtcfldext 31234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-mulr 16637  df-0g 16773  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-subg 18343  df-mgp 19308  df-ur 19320  df-ring 19367  df-subrg 19601  df-fldext 31238

This theorem is referenced by:  extdgmul  31257  finexttrb  31258