Theorem trirec0xor 15984

Description: Version of trirec0 15983 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 15983	. 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 9104	. . . . . . . 8 |- 1 =/= 0
3	2	nesymi 2422	. . . . . . 7 |- -. 0 = 1
4		simpr 110	. . . . . . . . . . 11 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> x = 0 )
5	4	oveq1d 5959	. . . . . . . . . 10 |- ( ( ( x x. z ) = 1 /\ x = 0 ) -> ( x x. z ) = ( 0 x. z ) )
6		mul02lem2 8460	. . . . . . . . . 10 |- ( z e. RR -> ( 0 x. z ) = 0 )
7	5, 6	sylan9eqr 2260	. . . . . . . . 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 2240	. . . . . . . 8 |- ( ( z e. RR /\ ( ( x x. z ) = 1 /\ x = 0 ) ) -> 0 = 1 )
10	9	rexlimiva 2618	. . . . . . 7 |- ( E. z e. RR ( ( x x. z ) = 1 /\ x = 0 ) -> 0 = 1 )
11	3, 10	mto 664	. . . . . 6 |- -. E. z e. RR ( ( x x. z ) = 1 /\ x = 0 )
12		r19.41v 2662	. . . . . 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 672	. . . . 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 1396	. . . 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 2512	. 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 710   \/ w3o 980   = wceq 1373   \/_ wxo 1395   e. wcel 2176   A.wral 2484   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   RRcr 7924   0cc0 7925   1c1 7926   x. cmul 7930   < clt 8107

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746

This theorem is referenced by: (None)