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Theorem trirec0 14448
Description: Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy.

This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 14447). (Contributed by Jim Kingdon, 10-Jun-2024.)

Assertion
Ref Expression
trirec0 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem trirec0
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
2 simpr 110 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 < 0)
31, 2lt0ap0d 8596 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 # 0)
4 rerecclap 8676 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℝ)
5 recn 7935 . . . . . . . 8 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
6 recidap 8632 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (𝑥 · (1 / 𝑥)) = 1)
75, 6sylan 283 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → (𝑥 · (1 / 𝑥)) = 1)
8 oveq2 5877 . . . . . . . . 9 (𝑧 = (1 / 𝑥) → (𝑥 · 𝑧) = (𝑥 · (1 / 𝑥)))
98eqeq1d 2186 . . . . . . . 8 (𝑧 = (1 / 𝑥) → ((𝑥 · 𝑧) = 1 ↔ (𝑥 · (1 / 𝑥)) = 1))
109rspcev 2841 . . . . . . 7 (((1 / 𝑥) ∈ ℝ ∧ (𝑥 · (1 / 𝑥)) = 1) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
114, 7, 10syl2anc 411 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
121, 3, 11syl2anc 411 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
1312orcd 733 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
14 simpr 110 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 = 0) → 𝑥 = 0)
1514olcd 734 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 = 0) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
16 simpll 527 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 𝑥 ∈ ℝ)
17 simpr 110 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 0 < 𝑥)
1816, 17gt0ap0d 8576 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 𝑥 # 0)
1916, 18, 11syl2anc 411 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
2019orcd 733 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
21 0re 7948 . . . . . 6 0 ∈ ℝ
22 breq2 4004 . . . . . . . 8 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
23 eqeq2 2187 . . . . . . . 8 (𝑦 = 0 → (𝑥 = 𝑦𝑥 = 0))
24 breq1 4003 . . . . . . . 8 (𝑦 = 0 → (𝑦 < 𝑥 ↔ 0 < 𝑥))
2522, 23, 243orbi123d 1311 . . . . . . 7 (𝑦 = 0 → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥)))
2625rspcv 2837 . . . . . 6 (0 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥)))
2721, 26ax-mp 5 . . . . 5 (∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥))
2827adantl 277 . . . 4 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) → (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥))
2913, 15, 20, 28mpjao3dan 1307 . . 3 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
3029ralimiaa 2539 . 2 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
31 oveq1 5876 . . . . . . 7 (𝑥 = 𝑤 → (𝑥 · 𝑧) = (𝑤 · 𝑧))
3231eqeq1d 2186 . . . . . 6 (𝑥 = 𝑤 → ((𝑥 · 𝑧) = 1 ↔ (𝑤 · 𝑧) = 1))
3332rexbidv 2478 . . . . 5 (𝑥 = 𝑤 → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ↔ ∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1))
34 eqeq1 2184 . . . . 5 (𝑥 = 𝑤 → (𝑥 = 0 ↔ 𝑤 = 0))
3533, 34orbi12d 793 . . . 4 (𝑥 = 𝑤 → ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)))
3635cbvralv 2703 . . 3 (∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ ∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0))
37 nfcv 2319 . . . . . . . . 9 𝑧
38 nfre1 2520 . . . . . . . . . 10 𝑧𝑧 ∈ ℝ (𝑤 · 𝑧) = 1
39 nfv 1528 . . . . . . . . . 10 𝑧 𝑤 = 0
4038, 39nfor 1574 . . . . . . . . 9 𝑧(∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)
4137, 40nfralya 2517 . . . . . . . 8 𝑧𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)
42 nfv 1528 . . . . . . . 8 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
4341, 42nfan 1565 . . . . . . 7 𝑧(∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
44 nfv 1528 . . . . . . 7 𝑧(𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)
45 simpr 110 . . . . . . . . . . 11 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → (𝑦𝑥) < 0)
46 simprr 531 . . . . . . . . . . . . . 14 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
4746ad2antrr 488 . . . . . . . . . . . . 13 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑦 ∈ ℝ)
4847adantr 276 . . . . . . . . . . . 12 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → 𝑦 ∈ ℝ)
49 simprl 529 . . . . . . . . . . . . . 14 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
5049ad2antrr 488 . . . . . . . . . . . . 13 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑥 ∈ ℝ)
5150adantr 276 . . . . . . . . . . . 12 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → 𝑥 ∈ ℝ)
5248, 51sublt0d 8517 . . . . . . . . . . 11 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → ((𝑦𝑥) < 0 ↔ 𝑦 < 𝑥))
5345, 52mpbid 147 . . . . . . . . . 10 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → 𝑦 < 𝑥)
54533mix3d 1174 . . . . . . . . 9 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
55 simpr 110 . . . . . . . . . . 11 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 0 < (𝑦𝑥))
5650adantr 276 . . . . . . . . . . . 12 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 𝑥 ∈ ℝ)
5747adantr 276 . . . . . . . . . . . 12 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 𝑦 ∈ ℝ)
5856, 57posdifd 8479 . . . . . . . . . . 11 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → (𝑥 < 𝑦 ↔ 0 < (𝑦𝑥)))
5955, 58mpbird 167 . . . . . . . . . 10 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 𝑥 < 𝑦)
60593mix1d 1172 . . . . . . . . 9 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
6147recnd 7976 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑦 ∈ ℂ)
6250recnd 7976 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑥 ∈ ℂ)
6361, 62subcld 8258 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) ∈ ℂ)
64 simplr 528 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑧 ∈ ℝ)
6564recnd 7976 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑧 ∈ ℂ)
66 simpr 110 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → ((𝑦𝑥) · 𝑧) = 1)
67 1ap0 8537 . . . . . . . . . . . 12 1 # 0
6866, 67eqbrtrdi 4039 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → ((𝑦𝑥) · 𝑧) # 0)
6963, 65, 68mulap0bad 8605 . . . . . . . . . 10 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) # 0)
7046, 49resubcld 8328 . . . . . . . . . . . 12 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑦𝑥) ∈ ℝ)
7170ad2antrr 488 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) ∈ ℝ)
72 reaplt 8535 . . . . . . . . . . 11 (((𝑦𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑦𝑥) # 0 ↔ ((𝑦𝑥) < 0 ∨ 0 < (𝑦𝑥))))
7371, 21, 72sylancl 413 . . . . . . . . . 10 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → ((𝑦𝑥) # 0 ↔ ((𝑦𝑥) < 0 ∨ 0 < (𝑦𝑥))))
7469, 73mpbid 147 . . . . . . . . 9 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → ((𝑦𝑥) < 0 ∨ 0 < (𝑦𝑥)))
7554, 60, 74mpjaodan 798 . . . . . . . 8 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
7675exp31 364 . . . . . . 7 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑧 ∈ ℝ → (((𝑦𝑥) · 𝑧) = 1 → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
7743, 44, 76rexlimd 2591 . . . . . 6 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
7877imp 124 . . . . 5 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ ∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
7946recnd 7976 . . . . . . . . 9 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
8079adantr 276 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑦 ∈ ℂ)
8149recnd 7976 . . . . . . . . 9 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℂ)
8281adantr 276 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑥 ∈ ℂ)
83 simpr 110 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → (𝑦𝑥) = 0)
8480, 82, 83subeq0d 8266 . . . . . . 7 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑦 = 𝑥)
8584equcomd 1707 . . . . . 6 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑥 = 𝑦)
86853mix2d 1173 . . . . 5 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
87 oveq1 5876 . . . . . . . . 9 (𝑤 = (𝑦𝑥) → (𝑤 · 𝑧) = ((𝑦𝑥) · 𝑧))
8887eqeq1d 2186 . . . . . . . 8 (𝑤 = (𝑦𝑥) → ((𝑤 · 𝑧) = 1 ↔ ((𝑦𝑥) · 𝑧) = 1))
8988rexbidv 2478 . . . . . . 7 (𝑤 = (𝑦𝑥) → (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ↔ ∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1))
90 eqeq1 2184 . . . . . . 7 (𝑤 = (𝑦𝑥) → (𝑤 = 0 ↔ (𝑦𝑥) = 0))
9189, 90orbi12d 793 . . . . . 6 (𝑤 = (𝑦𝑥) → ((∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ↔ (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 ∨ (𝑦𝑥) = 0)))
92 simpl 109 . . . . . 6 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0))
9391, 92, 70rspcdva 2846 . . . . 5 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 ∨ (𝑦𝑥) = 0))
9478, 86, 93mpjaodan 798 . . . 4 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9594ralrimivva 2559 . . 3 (∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9636, 95sylbi 121 . 2 (∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9730, 96impbii 126 1 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3o 977   = wceq 1353  wcel 2148  wral 2455  wrex 2456   class class class wbr 4000  (class class class)co 5869  cc 7800  cr 7801  0cc0 7802  1c1 7803   · cmul 7807   < clt 7982  cmin 8118   # cap 8528   / cdiv 8618
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619
This theorem is referenced by:  trirec0xor  14449
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