Theorem trirec0xor 15689

Description: Version of trirec0 15688 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 15688	. 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 9058	. . . . . . . 8 |- 1 =/= 0
3	2	nesymi 2413	. . . . . . 7 |- -. 0 = 1
4		simpr 110	. . . . . . . . . . 11 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> x = 0 )
5	4	oveq1d 5937	. . . . . . . . . 10 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> ( x x. z ) = ( 0 x. z ) )
6		mul02lem2 8414	. . . . . . . . . 10 |- ( z e. RR -> ( 0 x. z ) = 0 )
7	5, 6	sylan9eqr 2251	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 0 )
8		simprl 529	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> ( x x. z ) = 1 )
9	7, 8	eqtr3d 2231	. . . . . . . 8 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> 0 = 1 )
10	9	rexlimiva 2609	. . . . . . 7 |- ( E. z e. RR ( ( x x. z ) = 1 /\ x = 0 ) -> 0 = 1 )
11	3, 10	mto 663	. . . . . 6 |- -. E. z e. RR ( ( x x. z ) = 1 /\ x = 0 )
12		r19.41v 2653	. . . . . 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 671	. . . . 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 1387	. . . 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 2503	. 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 709   \/ w3o 979   = wceq 1364   \/_ wxo 1386   e. wcel 2167   A.wral 2475   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   x. cmul 7884   < clt 8061

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700

This theorem is referenced by: (None)