Theorem trirec0xor 16955

Description: Version of trirec0 16954 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 16954	. 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 9322	. . . . . . . 8 |- 1 =/= 0
3	2	nesymi 2460	. . . . . . 7 |- -. 0 = 1
4		simpr 110	. . . . . . . . . . 11 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> x = 0 )
5	4	oveq1d 6073	. . . . . . . . . 10 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> ( x x. z ) = ( 0 x. z ) )
6		mul02lem2 8678	. . . . . . . . . 10 |- ( z e. RR -> ( 0 x. z ) = 0 )
7	5, 6	sylan9eqr 2289	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 0 )
8		simprl 531	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 1 )
9	7, 8	eqtr3d 2269	. . . . . . . 8 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> 0 = 1 )
10	9	rexlimiva 2657	. . . . . . 7 |- ( E. z e. RR ( ( x x. z ) = 1 /\ x = 0 ) -> 0 = 1 )
11	3, 10	mto 668	. . . . . 6 |- -. E. z e. RR ( ( x x. z ) = 1 /\ x = 0 )
12		r19.41v 2701	. . . . . 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 677	. . . . 5 |- -. ( E. z e. RR ( x x. z ) = 1 /\ x = 0 )
14	13	biantru 302	. . . 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 1421	. . . 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 187	. . 3 |- ( ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) <-> ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )
17	16	ralbii 2550	. 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 184	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 104   <-> wb 105   \/ wo 716   \/ w3o 1004   = wceq 1398   \/_ wxo 1420   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324

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-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964

This theorem is referenced by: (None)