Theorem fldexttr 34290

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 490	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2		simpl 488	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3		fldextfld2 34280	. . . . . . 7 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4	2, 3	syl 18	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5		fldextfld2 34280	. . . . . . 7 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
6	1, 5	syl 18	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7		brfldext 34277	. . . . . 6 ⊢ ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
8	4, 6, 7	syl2anc 596	. . . . 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 500	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐹 ↾s (Base‘𝐾)))
11		fldextfld1 34279	. . . . . . . . 9 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
12	2, 11	syl 18	. . . . . . . 8 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13		brfldext 34277	. . . . . . . 8 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
14	12, 4, 13	syl2anc 596	. . . . . . 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 500	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
17	16	oveq1d 7435	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)))
18		fvex 6898	. . . . 5 ⊢ (Base‘𝐹) ∈ V
19		fvex 6898	. . . . 5 ⊢ (Base‘𝐾) ∈ V
20		ressress 17425	. . . . 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 705	. . . 4 ⊢ ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾)))
22	17, 21	eqtrdi 2812	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))))
23		incom 4155	. . . . 5 ⊢ ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
24	9	simprd 501	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	subrgss 20824	. . . . . . 7 ⊢ ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
27	24, 26	syl 18	. . . . . 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 221	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
30	23, 29	eqtr3id 2810	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
31	30	oveq2d 7436	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸 ↾s (Base‘𝐾)))
32	10, 22, 31	3eqtrd 2800	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
33	15	simprd 501	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34	16	fveq2d 6889	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
35	24, 34	eleqtrd 2863	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
36		eqid 2761	. . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
37	36	subsubrg 20850	. . . 4 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
38	37	simprbda 504	. . 3 ⊢ (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
39	33, 35, 38	syl2anc 596	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40		brfldext 34277	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
41	12, 6, 40	syl2anc 596	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
42	32, 39, 41	mpbir2and 726	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387   ↾_s cress 17408  SubRingcsubrg 20821  Fieldcfield 20981  /_FldExtcfldext 34270

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-subg 19333  df-mgp 20361  df-ur 20408  df-ring 20461  df-subrg 20822  df-fldext 34273

This theorem is used by:  extdgmul  34295  finexttrb  34297  fldext2chn  34360