Theorem fldexttr 33802

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 33792	. . . . . . 7 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5		fldextfld2 33792	. . . . . . 7 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
6	1, 5	syl 17	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7		brfldext 33789	. . . . . 6 ⊢ ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
8	4, 6, 7	syl2anc 585	. . . . 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 33791	. . . . . . . . 9 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
12	2, 11	syl 17	. . . . . . . 8 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13		brfldext 33789	. . . . . . . 8 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
14	12, 4, 13	syl2anc 585	. . . . . . 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 7382	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 ↾s (Base‘𝐾)) = ((𝐸 ↾s (Base‘𝐹)) ↾s (Base‘𝐾)))
18		fvex 6853	. . . . 5 ⊢ (Base‘𝐹) ∈ V
19		fvex 6853	. . . . 5 ⊢ (Base‘𝐾) ∈ V
20		ressress 17217	. . . . 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 693	. . . 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 4149	. . . . 5 ⊢ ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
24	9	simprd 495	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	subrgss 20549	. . . . . . 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 3907	. . . . . 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 2785	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
31	30	oveq2d 7383	. . 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 495	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34	16	fveq2d 6844	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
35	24, 34	eleqtrd 2838	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
36		eqid 2736	. . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
37	36	subsubrg 20575	. . . 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 585	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40		brfldext 33789	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
41	12, 6, 40	syl2anc 585	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
42	32, 39, 41	mpbir2and 714	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  SubRingcsubrg 20546  Fieldcfield 20707  /_FldExtcfldext 33782

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-subg 19099  df-mgp 20122  df-ur 20163  df-ring 20216  df-subrg 20547  df-fldext 33785

This theorem is referenced by:  extdgmul  33807  finexttrb  33809  fldext2chn  33872