Theorem isdivrngNEW 17160

Description: The predicate "is a division ring".

Hypotheses

Ref	Expression
isdivrng.1NEW	|- B = (base` R)
isdivrng.3NEW	|- T = (.r` R)
isdivrng.4NEW	|- Z = (0g` R)
isdivrng.6NEW	|- G = {<.1, (B \ {Z})>., <.2, T>.}

Assertion

Ref	Expression
isdivrngNEW	|- (R e. DivRingNEW <-> (R e. RingNEW /\ G e. GrpNEW))

Proof of Theorem isdivrngNEW

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . . . . 10 |- (r = R -> (base` r) = (base` R))
2		isdivrng.1NEW	. . . . . . . . . 10 |- B = (base` R)
3	1, 2	syl6eqr 1946	. . . . . . . . 9 |- (r = R -> (base` r) = B)
4	3	difeq1d 2725	. . . . . . . 8 |- (r = R -> ((base` r) \ {(0g` r)}) = (B \ {(0g` r)}))
5		fveq2 4681	. . . . . . . . . . 11 |- (r = R -> (0g` r) = (0g` R))
6	5	sneqd 3056	. . . . . . . . . 10 |- (r = R -> {(0g` r)} = {(0g` R)})
7		isdivrng.4NEW	. . . . . . . . . . 11 |- Z = (0g` R)
8	7	sneqi 3055	. . . . . . . . . 10 |- {Z} = {(0g` R)}
9	6, 8	syl6eqr 1946	. . . . . . . . 9 |- (r = R -> {(0g` r)} = {Z})
10	9	difeq2d 2726	. . . . . . . 8 |- (r = R -> (B \ {(0g` r)}) = (B \ {Z}))
11	4, 10	eqtrd 1925	. . . . . . 7 |- (r = R -> ((base` r) \ {(0g` r)}) = (B \ {Z}))
12	11	opeq2d 3165	. . . . . 6 |- (r = R -> <.1, ((base` r) \ {(0g` r)})>. = <.1, (B \ {Z})>.)
13		preq1 3098	. . . . . 6 |- (<.1, ((base` r) \ {(0g` r)})>. = <.1, (B \ {Z})>. -> {<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} = {<.1, (B \ {Z})>., <.2, (.r` r)>.})
14	12, 13	syl 12	. . . . 5 |- (r = R -> {<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} = {<.1, (B \ {Z})>., <.2, (.r` r)>.})
15		fveq2 4681	. . . . . . . 8 |- (r = R -> (.r` r) = (.r` R))
16		isdivrng.3NEW	. . . . . . . 8 |- T = (.r` R)
17	15, 16	syl6eqr 1946	. . . . . . 7 |- (r = R -> (.r` r) = T)
18	17	opeq2d 3165	. . . . . 6 |- (r = R -> <.2, (.r` r)>. = <.2, T>.)
19		preq2 3099	. . . . . 6 |- (<.2, (.r` r)>. = <.2, T>. -> {<.1, (B \ {Z})>., <.2, (.r` r)>.} = {<.1, (B \ {Z})>., <.2, T>.})
20	18, 19	syl 12	. . . . 5 |- (r = R -> {<.1, (B \ {Z})>., <.2, (.r` r)>.} = {<.1, (B \ {Z})>., <.2, T>.})
21	14, 20	eqtrd 1925	. . . 4 |- (r = R -> {<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} = {<.1, (B \ {Z})>., <.2, T>.})
22	21	eleq1d 1963	. . 3 |- (r = R -> ({<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} e. GrpNEW <-> {<.1, (B \ {Z})>., <.2, T>.} e. GrpNEW))
23		isdivrng.6NEW	. . . 4 |- G = {<.1, (B \ {Z})>., <.2, T>.}
24	23	eleq1i 1960	. . 3 |- (G e. GrpNEW <-> {<.1, (B \ {Z})>., <.2, T>.} e. GrpNEW)
25	22, 24	syl6bbr 597	. 2 |- (r = R -> ({<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} e. GrpNEW <-> G e. GrpNEW))
26		df-drngNEW 17159	. 2 |- DivRingNEW = {r e. RingNEW | {<.1, ((base` r) \ {(0g` r)})>., <.2, (.r` r)>.} e. GrpNEW}
27	25, 26	elrab2 2416	1 |- (R e. DivRingNEW <-> (R e. RingNEW /\ G e. GrpNEW))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590  {csn 3044  {cpr 3045  <.cop 3046  ` cfv 3998  1c1 6387  2c2 7145  basecbs 16758  GrpNEWcgrp 17081  0_gc0g 17082  ._rcmulr 17085  RingNEWcrg 17086  DivRingNEWcdivring 17158

This theorem is referenced by:  divrngring 17161  divrngmgrpNEW 17163  divrngidlemNEW 17165

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-drngNEW 17159

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