Theorem fldexttr 32575

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 485	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2		simpl 483	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3		fldextfld2 32567	. . . . . . 7 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5		fldextfld2 32567	. . . . . . 7 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
6	1, 5	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7		brfldext 32564	. . . . . 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 231	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹)))
10	9	simpld 495	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐹 ↾s (Base‘𝐾)))
11		fldextfld1 32566	. . . . . . . . 9 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
12	2, 11	syl 17	. . . . . . . 8 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13		brfldext 32564	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
16	15	simpld 495	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
17	16	oveq1d 7408	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)))
18		fvex 6891	. . . . 5 ⊢ (Base‘𝐹) ∈ V
19		fvex 6891	. . . . 5 ⊢ (Base‘𝐾) ∈ V
20		ressress 17175	. . . . 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 690	. . . 4 ⊢ ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾)))
22	17, 21	eqtrdi 2787	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))))
23		incom 4197	. . . . 5 ⊢ ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
24	9	simprd 496	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	subrgss 20313	. . . . . . 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 3961	. . . . . 6 ⊢ ((Base‘𝐾) ⊆ (Base‘𝐹) ↔ ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
29	27, 28	sylib 217	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
30	23, 29	eqtr3id 2785	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
31	30	oveq2d 7409	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸 ↾s (Base‘𝐾)))
32	10, 22, 31	3eqtrd 2775	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
33	15	simprd 496	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34	16	fveq2d 6882	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
35	24, 34	eleqtrd 2834	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
36		eqid 2731	. . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
37	36	subsubrg 20339	. . . 4 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
38	37	simprbda 499	. . 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 32564	. . 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 711	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3473   ∩ cin 3943   ⊆ wss 3944   class class class wbr 5141  ‘cfv 6532  (class class class)co 7393  Basecbs 17126   ↾_s cress 17155  Fieldcfield 20266  SubRingcsubrg 20308  /_FldExtcfldext 32555

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-subg 18975  df-mgp 19947  df-ur 19964  df-ring 20016  df-subrg 20310  df-fldext 32559

This theorem is referenced by:  extdgmul  32578  finexttrb  32579