Theorem trirec0xor 13924

Description: Version of trirec0 13923 with exclusive-or.

The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)

Assertion

Ref	Expression
trirec0xor	|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )

Distinct variable group:   x, y, z

Proof of Theorem trirec0xor

Step	Hyp	Ref	Expression
1		trirec0 13923	. 2 |- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) )
2		1ne0 8925	. . . . . . . 8 |- 1 =/= 0
3	2	nesymi 2382	. . . . . . 7 |- -. 0 = 1
4		simpr 109	. . . . . . . . . . 11 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> x = 0 )
5	4	oveq1d 5857	. . . . . . . . . 10 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> ( x x. z ) = ( 0 x. z ) )
6		mul02lem2 8286	. . . . . . . . . 10 |- ( z e. RR -> ( 0 x. z ) = 0 )
7	5, 6	sylan9eqr 2221	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 0 )
8		simprl 521	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 1 )
9	7, 8	eqtr3d 2200	. . . . . . . 8 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> 0 = 1 )
10	9	rexlimiva 2578	. . . . . . 7 |- ( E. z e. RR ( ( x x. z ) = 1 /\ x = 0 ) -> 0 = 1 )
11	3, 10	mto 652	. . . . . 6 |- -. E. z e. RR ( ( x x. z ) = 1 /\ x = 0 )
12		r19.41v 2622	. . . . . 6 |- ( E. z e. RR ( ( x x. z ) = 1 /\ x = 0 ) <-> ( E. z e. RR ( x x. z ) = 1 /\ x = 0 ) )
13	11, 12	mtbi 660	. . . . 5 |- -. ( E. z e. RR ( x x. z ) = 1 /\ x = 0 )
14	13	biantru 300	. . . 4 |- ( ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) <-> ( ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) /\ -. ( E. z e. RR ( x x. z ) = 1 /\ x = 0 ) ) )
15		df-xor 1366	. . . 4 |- ( ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) <-> ( ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) /\ -. ( E. z e. RR ( x x. z ) = 1 /\ x = 0 ) ) )
16	14, 15	bitr4i 186	. . 3 |- ( ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) <-> ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )
17	16	ralbii 2472	. 2 |- ( A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )
18	1, 17	bitri 183	1 |- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   \/ wo 698   \/ w3o 967   = wceq 1343   \/_ wxo 1365   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   x. cmul 7758   < clt 7933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569

This theorem is referenced by: (None)