Theorem trirec0xor 13758

Description: Version of trirec0 13757 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 13757	. 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 8916	. . . . . . . 8 |- 1 =/= 0
3	2	nesymi 2380	. . . . . . 7 |- -. 0 = 1
4		simpr 109	. . . . . . . . . . 11 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> x = 0 )
5	4	oveq1d 5851	. . . . . . . . . 10 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> ( x x. z ) = ( 0 x. z ) )
6		mul02lem2 8277	. . . . . . . . . 10 |- ( z e. RR -> ( 0 x. z ) = 0 )
7	5, 6	sylan9eqr 2219	. . . . . . . . 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 2199	. . . . . . . 8 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> 0 = 1 )
10	9	rexlimiva 2576	. . . . . . 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 2620	. . . . . 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 1365	. . . 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 2470	. 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 966   = wceq 1342   \/_ wxo 1364   e. wcel 2135   A.wral 2442   E.wrex 2443   class class class wbr 3976  (class class class)co 5836   RRcr 7743   0cc0 7744   1c1 7745   x. cmul 7749   < clt 7924

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-xor 1365  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-po 4268  df-iso 4269  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560

This theorem is referenced by: (None)