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Theorem fldexttr 32404
Description: Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
fldexttr ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Proof of Theorem fldexttr
StepHypRef Expression
1 simpr 486 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2 simpl 484 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3 fldextfld2 32396 . . . . . . 7 (𝐸/FldExt𝐹𝐹 ∈ Field)
42, 3syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5 fldextfld2 32396 . . . . . . 7 (𝐹/FldExt𝐾𝐾 ∈ Field)
61, 5syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7 brfldext 32393 . . . . . 6 ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
84, 6, 7syl2anc 585 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
91, 8mpbid 231 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹)))
109simpld 496 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐹s (Base‘𝐾)))
11 fldextfld1 32395 . . . . . . . . 9 (𝐸/FldExt𝐹𝐸 ∈ Field)
122, 11syl 17 . . . . . . . 8 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13 brfldext 32393 . . . . . . . 8 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
1412, 4, 13syl2anc 585 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
152, 14mpbid 231 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
1615simpld 496 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 = (𝐸s (Base‘𝐹)))
1716oveq1d 7373 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)))
18 fvex 6856 . . . . 5 (Base‘𝐹) ∈ V
19 fvex 6856 . . . . 5 (Base‘𝐾) ∈ V
20 ressress 17134 . . . . 5 (((Base‘𝐹) ∈ V ∧ (Base‘𝐾) ∈ V) → ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
2118, 19, 20mp2an 691 . . . 4 ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾)))
2217, 21eqtrdi 2789 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
23 incom 4162 . . . . 5 ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
249simprd 497 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25 eqid 2733 . . . . . . . 8 (Base‘𝐹) = (Base‘𝐹)
2625subrgss 20237 . . . . . . 7 ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
2724, 26syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ⊆ (Base‘𝐹))
28 df-ss 3928 . . . . . 6 ((Base‘𝐾) ⊆ (Base‘𝐹) ↔ ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
2927, 28sylib 217 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
3023, 29eqtr3id 2787 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
3130oveq2d 7374 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸s (Base‘𝐾)))
3210, 22, 313eqtrd 2777 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐸s (Base‘𝐾)))
3315simprd 497 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
3416fveq2d 6847 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3524, 34eleqtrd 2836 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
36 eqid 2733 . . . . 5 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
3736subsubrg 20262 . . . 4 ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
3837simprbda 500 . . 3 (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
3933, 35, 38syl2anc 585 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40 brfldext 32393 . . 3 ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4112, 6, 40syl2anc 585 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4232, 39, 41mpbir2and 712 1 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  cin 3910  wss 3911   class class class wbr 5106  cfv 6497  (class class class)co 7358  Basecbs 17088  s cress 17117  Fieldcfield 20198  SubRingcsubrg 20232  /FldExtcfldext 32384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-subg 18930  df-mgp 19902  df-ur 19919  df-ring 19971  df-subrg 20234  df-fldext 32388
This theorem is referenced by:  extdgmul  32407  finexttrb  32408
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