Theorem fldi 14776

Description: The "axioms" of a field.

Hypotheses

Ref	Expression
fldi.1	|- G = (1st` R)
fldi.2	|- H = (2nd` R)
fldi.3	|- X = ran G
fldi.4	|- Z = (Id` G)

Assertion

Ref	Expression
fldi	|- (R e. Fld -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))))

Distinct variable groups:   x,G,y,z   x,H,y,z   x,R,y   x,X,y,z

Proof of Theorem fldi

Step	Hyp	Ref	Expression
1		df-fld 10398	. . 3 |- Fld = (DivRing i^i Com2)
2	1	eleq2i 1961	. 2 |- (R e. Fld <-> R e. (DivRing i^i Com2))
3		elin 2786	. . 3 |- (R e. (DivRing i^i Com2) <-> (R e. DivRing /\ R e. Com2))
4		fldi.1	. . . . . 6 |- G = (1st` R)
5		fldi.2	. . . . . 6 |- H = (2nd` R)
6		fldi.3	. . . . . 6 |- X = ran G
7		fldi.4	. . . . . 6 |- Z = (Id` G)
8	4, 5, 6, 7	drngi 9493	. . . . 5 |- (R e. DivRing -> (R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
9	4, 5, 6	ringi 9466	. . . . . . . 8 |- (R e. Ring -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))))
10		id 73	. . . . . . . . 9 |- (((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))))
11	10	3expia 1069	. . . . . . . 8 |- (((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))) -> (((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)))))
12	9, 11	syl 12	. . . . . . 7 |- (R e. Ring -> (((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)))))
13	12	expdimp 406	. . . . . 6 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> (A.x e. X A.y e. X (xHy) = (yHx) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)))))
14	4, 5, 6	com2i 14765	. . . . . 6 |- (R e. Com2 -> A.x e. X A.y e. X (xHy) = (yHx))
15	13, 14	syl5 20	. . . . 5 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> (R e. Com2 -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)))))
16	8, 15	syl 12	. . . 4 |- (R e. DivRing -> (R e. Com2 -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx)))))
17	16	imp 377	. . 3 |- ((R e. DivRing /\ R e. Com2) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))))
18	3, 17	sylbi 216	. 2 |- (R e. (DivRing i^i Com2) -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))))
19	2, 18	sylbi 216	1 |- (R e. Fld -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)) /\ ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A.x e. X A.y e. X (xHy) = (yHx))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   \ cdif 2590   i^i cin 2592  {csn 3044   X. cxp 3984  ran crn 3987   |` cres 3988  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  Abelcabl 9407  Ringcring 9463  DivRingcdrng 9491  Com2ccm2 10394  Fldcfld 10397

This theorem is referenced by:  fldax1 14777  fldax2 14778  fldax3 14779  fldax4 14780  fldax5 14781  fldax6 14782  fldax7 14783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464  df-drng 9492  df-com2 10395  df-fld 10398

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