Theorem isdrngtap 14466

Description: The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)

Hypotheses

Ref	Expression
isdrng.b	|- B = ( Base ` R )
isdrngap.ap	|- # = (#r ` R )

Assertion

Ref	Expression
isdrngtap	|- ( R e. DivRing <-> ( R e. Ring /\ # TAp B ) )

Proof of Theorem isdrngtap

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5672	. . . . 5 |- ( r = R -> (#r ` r ) = (#r ` R ) )
2		isdrngap.ap	. . . . 5 |- # = (#r ` R )
3	1, 2	eqtr4di 2285	. . . 4 |- ( r = R -> (#r ` r ) = # )
4		tapeq1 7571	. . . 4 |- ( (#r ` r ) = # -> ( (#r ` r ) TAp ( Base ` r ) <-> # TAp ( Base ` r ) ) )
5	3, 4	syl 14	. . 3 |- ( r = R -> ( (#r ` r ) TAp ( Base ` r ) <-> # TAp ( Base ` r ) ) )
6		fveq2 5672	. . . . 5 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
7		isdrng.b	. . . . 5 |- B = ( Base ` R )
8	6, 7	eqtr4di 2285	. . . 4 |- ( r = R -> ( Base ` r ) = B )
9		tapeq2 7572	. . . 4 |- ( ( Base ` r ) = B -> ( # TAp ( Base ` r ) <-> # TAp B ) )
10	8, 9	syl 14	. . 3 |- ( r = R -> ( # TAp ( Base ` r ) <-> # TAp B ) )
11	5, 10	bitrd 188	. 2 |- ( r = R -> ( (#r ` r ) TAp ( Base ` r ) <-> # TAp B ) )
12		df-drngap 14464	. 2 |- DivRing = { r e. Ring | (#r ` r ) TAp ( Base ` r ) }
13	11, 12	elrab2 2978	1 |- ( R e. DivRing <-> ( R e. Ring /\ # TAp B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   ` cfv 5354   TAp wtap 7567   Basecbs 13233   Ringcrg 14161  #_rcapr 14449   DivRingcdr 14462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-iota 5314  df-fv 5362  df-pap 7561  df-tap 7568  df-drngap 14464

This theorem is referenced by:  drnglring  14467  drngprop  14477  opprdrng  14480