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Theorem trirec0 13412
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 13411). (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 519 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
2 simpr 109 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 < 0)
31, 2lt0ap0d 8435 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → 𝑥 # 0)
4 rerecclap 8514 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℝ)
5 recn 7777 . . . . . . . 8 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
6 recidap 8470 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (𝑥 · (1 / 𝑥)) = 1)
75, 6sylan 281 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → (𝑥 · (1 / 𝑥)) = 1)
8 oveq2 5790 . . . . . . . . 9 (𝑧 = (1 / 𝑥) → (𝑥 · 𝑧) = (𝑥 · (1 / 𝑥)))
98eqeq1d 2149 . . . . . . . 8 (𝑧 = (1 / 𝑥) → ((𝑥 · 𝑧) = 1 ↔ (𝑥 · (1 / 𝑥)) = 1))
109rspcev 2793 . . . . . . 7 (((1 / 𝑥) ∈ ℝ ∧ (𝑥 · (1 / 𝑥)) = 1) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
114, 7, 10syl2anc 409 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑥 # 0) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
121, 3, 11syl2anc 409 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
1312orcd 723 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 < 0) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
14 simpr 109 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 = 0) → 𝑥 = 0)
1514olcd 724 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 𝑥 = 0) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
16 simpll 519 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 𝑥 ∈ ℝ)
17 simpr 109 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 0 < 𝑥)
1816, 17gt0ap0d 8415 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → 𝑥 # 0)
1916, 18, 11syl2anc 409 . . . . 5 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → ∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1)
2019orcd 723 . . . 4 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ 0 < 𝑥) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
21 0re 7790 . . . . . 6 0 ∈ ℝ
22 breq2 3941 . . . . . . . 8 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
23 eqeq2 2150 . . . . . . . 8 (𝑦 = 0 → (𝑥 = 𝑦𝑥 = 0))
24 breq1 3940 . . . . . . . 8 (𝑦 = 0 → (𝑦 < 𝑥 ↔ 0 < 𝑥))
2522, 23, 243orbi123d 1290 . . . . . . 7 (𝑦 = 0 → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥)))
2625rspcv 2789 . . . . . 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 275 . . . 4 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) → (𝑥 < 0 ∨ 𝑥 = 0 ∨ 0 < 𝑥))
2913, 15, 20, 28mpjao3dan 1286 . . 3 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
3029ralimiaa 2497 . 2 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
31 oveq1 5789 . . . . . . 7 (𝑥 = 𝑤 → (𝑥 · 𝑧) = (𝑤 · 𝑧))
3231eqeq1d 2149 . . . . . 6 (𝑥 = 𝑤 → ((𝑥 · 𝑧) = 1 ↔ (𝑤 · 𝑧) = 1))
3332rexbidv 2439 . . . . 5 (𝑥 = 𝑤 → (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ↔ ∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1))
34 eqeq1 2147 . . . . 5 (𝑥 = 𝑤 → (𝑥 = 0 ↔ 𝑤 = 0))
3533, 34orbi12d 783 . . . 4 (𝑥 = 𝑤 → ((∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)))
3635cbvralv 2657 . . 3 (∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) ↔ ∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0))
37 nfcv 2282 . . . . . . . . 9 𝑧
38 nfre1 2479 . . . . . . . . . 10 𝑧𝑧 ∈ ℝ (𝑤 · 𝑧) = 1
39 nfv 1509 . . . . . . . . . 10 𝑧 𝑤 = 0
4038, 39nfor 1554 . . . . . . . . 9 𝑧(∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)
4137, 40nfralya 2476 . . . . . . . 8 𝑧𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0)
42 nfv 1509 . . . . . . . 8 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
4341, 42nfan 1545 . . . . . . 7 𝑧(∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
44 nfv 1509 . . . . . . 7 𝑧(𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)
45 simpr 109 . . . . . . . . . . 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 522 . . . . . . . . . . . . . 14 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
4746ad2antrr 480 . . . . . . . . . . . . 13 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑦 ∈ ℝ)
4847adantr 274 . . . . . . . . . . . 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 521 . . . . . . . . . . . . . 14 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
5049ad2antrr 480 . . . . . . . . . . . . 13 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑥 ∈ ℝ)
5150adantr 274 . . . . . . . . . . . 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 8356 . . . . . . . . . . 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 146 . . . . . . . . . 10 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ (𝑦𝑥) < 0) → 𝑦 < 𝑥)
54533mix3d 1159 . . . . . . . . 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 109 . . . . . . . . . . 11 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 0 < (𝑦𝑥))
5650adantr 274 . . . . . . . . . . . 12 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 𝑥 ∈ ℝ)
5747adantr 274 . . . . . . . . . . . 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 8318 . . . . . . . . . . 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 166 . . . . . . . . . 10 (((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) ∧ 0 < (𝑦𝑥)) → 𝑥 < 𝑦)
60593mix1d 1157 . . . . . . . . 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 7818 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑦 ∈ ℂ)
6250recnd 7818 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑥 ∈ ℂ)
6361, 62subcld 8097 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) ∈ ℂ)
64 simplr 520 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑧 ∈ ℝ)
6564recnd 7818 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → 𝑧 ∈ ℂ)
66 simpr 109 . . . . . . . . . . . 12 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → ((𝑦𝑥) · 𝑧) = 1)
67 1ap0 8376 . . . . . . . . . . . 12 1 # 0
6866, 67eqbrtrdi 3975 . . . . . . . . . . 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 8444 . . . . . . . . . 10 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) # 0)
7046, 49resubcld 8167 . . . . . . . . . . . 12 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑦𝑥) ∈ ℝ)
7170ad2antrr 480 . . . . . . . . . . 11 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑦𝑥) ∈ ℝ)
72 reaplt 8374 . . . . . . . . . . 11 (((𝑦𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑦𝑥) # 0 ↔ ((𝑦𝑥) < 0 ∨ 0 < (𝑦𝑥))))
7371, 21, 72sylancl 410 . . . . . . . . . 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 146 . . . . . . . . 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 788 . . . . . . . 8 ((((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑦𝑥) · 𝑧) = 1) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
7675exp31 362 . . . . . . 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 2549 . . . . . 6 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
7877imp 123 . . . . 5 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ ∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
7946recnd 7818 . . . . . . . . 9 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
8079adantr 274 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑦 ∈ ℂ)
8149recnd 7818 . . . . . . . . 9 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℂ)
8281adantr 274 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑥 ∈ ℂ)
83 simpr 109 . . . . . . . 8 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → (𝑦𝑥) = 0)
8480, 82, 83subeq0d 8105 . . . . . . 7 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑦 = 𝑥)
8584equcomd 1684 . . . . . 6 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → 𝑥 = 𝑦)
86853mix2d 1158 . . . . 5 (((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (𝑦𝑥) = 0) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
87 oveq1 5789 . . . . . . . . 9 (𝑤 = (𝑦𝑥) → (𝑤 · 𝑧) = ((𝑦𝑥) · 𝑧))
8887eqeq1d 2149 . . . . . . . 8 (𝑤 = (𝑦𝑥) → ((𝑤 · 𝑧) = 1 ↔ ((𝑦𝑥) · 𝑧) = 1))
8988rexbidv 2439 . . . . . . 7 (𝑤 = (𝑦𝑥) → (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ↔ ∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1))
90 eqeq1 2147 . . . . . . 7 (𝑤 = (𝑦𝑥) → (𝑤 = 0 ↔ (𝑦𝑥) = 0))
9189, 90orbi12d 783 . . . . . 6 (𝑤 = (𝑦𝑥) → ((∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ↔ (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 ∨ (𝑦𝑥) = 0)))
92 simpl 108 . . . . . 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 2798 . . . . 5 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∃𝑧 ∈ ℝ ((𝑦𝑥) · 𝑧) = 1 ∨ (𝑦𝑥) = 0))
9478, 86, 93mpjaodan 788 . . . 4 ((∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9594ralrimivva 2517 . . 3 (∀𝑤 ∈ ℝ (∃𝑧 ∈ ℝ (𝑤 · 𝑧) = 1 ∨ 𝑤 = 0) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9636, 95sylbi 120 . 2 (∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
9730, 96impbii 125 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 103  wb 104  wo 698  w3o 962   = wceq 1332  wcel 1481  wral 2417  wrex 2418   class class class wbr 3937  (class class class)co 5782  cc 7642  cr 7643  0cc0 7644  1c1 7645   · cmul 7649   < clt 7824  cmin 7957   # cap 8367   / cdiv 8456
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457
This theorem is referenced by:  trirec0xor  13413
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