Theorem trirec0xor 16443

Description: Version of trirec0 16442 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	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem trirec0xor

Step	Hyp	Ref	Expression
1		trirec0 16442	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
2		1ne0 9186	. . . . . . . 8 ⊢ 1 ≠ 0
3	2	nesymi 2446	. . . . . . 7 ⊢ ¬ 0 = 1
4		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0) → 𝑥 = 0)
5	4	oveq1d 6022	. . . . . . . . . 10 ⊢ (((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0) → (𝑥 · 𝑧) = (0 · 𝑧))
6		mul02lem2 8542	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → (0 · 𝑧) = 0)
7	5, 6	sylan9eqr 2284	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)) → (𝑥 · 𝑧) = 0)
8		simprl 529	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)) → (𝑥 · 𝑧) = 1)
9	7, 8	eqtr3d 2264	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ ∧ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)) → 0 = 1)
10	9	rexlimiva 2643	. . . . . . 7 ⊢ (∃𝑧 ∈ ℝ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0) → 0 = 1)
11	3, 10	mto 666	. . . . . 6 ⊢ ¬ ∃𝑧 ∈ ℝ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)
12		r19.41v 2687	. . . . . 6 ⊢ (∃𝑧 ∈ ℝ ((𝑥 · 𝑧) = 1 ∧ 𝑥 = 0) ↔ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∧ 𝑥 = 0))
13	11, 12	mtbi 674	. . . . 5 ⊢ ¬ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)
14	13	biantru 302	. . . 4 ⊢ ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ∧ ¬ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)))
15		df-xor 1418	. . . 4 ⊢ ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0) ↔ ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ∧ ¬ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∧ 𝑥 = 0)))
16	14, 15	bitr4i 187	. . 3 ⊢ ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))
17	16	ralbii 2536	. 2 ⊢ (∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))
18	1, 17	bitri 184	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∨ w3o 1001   = wceq 1395   ⊻ wxo 1417   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   class class class wbr 4083  (class class class)co 6007  ℝcr 8006  0cc0 8007  1c1 8008   · cmul 8012   < clt 8189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828

This theorem is referenced by: (None)