Theorem ltsor 4055

Description: 'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 4279 and not this one after the complex number postulates are derived, in order to maintain a "clean" derivation of complex number theorems directly from postulates.

Assertion

Ref	Expression
ltsor	⊢ < Or ℝ

Proof of Theorem ltsor

Step	Hyp	Ref	Expression
1		elreal 4044	. . 3 ⊢ (x ∈ ℝ ↔ ∃w(w ∈ R ∧ ⟨w, 0R⟩ = x))
2		elreal 4044	. . 3 ⊢ (y ∈ ℝ ↔ ∃v(v ∈ R ∧ ⟨v, 0R⟩ = y))
3		elreal 4044	. . 3 ⊢ (z ∈ ℝ ↔ ∃u(u ∈ R ∧ ⟨u, 0R⟩ = z))
4		breq1 2065	. . . . 5 ⊢ (⟨w, 0R⟩ = x → (⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ x < ⟨v, 0R⟩))
5		cleq1 1107	. . . . . . 7 ⊢ (⟨w, 0R⟩ = x → (⟨w, 0R⟩ = ⟨v, 0R⟩ ↔ x = ⟨v, 0R⟩))
6		breq2 2066	. . . . . . 7 ⊢ (⟨w, 0R⟩ = x → (⟨v, 0R⟩ < ⟨w, 0R⟩ ↔ ⟨v, 0R⟩ < x))
7	5, 6	orbi12d 475	. . . . . 6 ⊢ (⟨w, 0R⟩ = x → ((⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩) ↔ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x)))
8	7	negbid 463	. . . . 5 ⊢ (⟨w, 0R⟩ = x → (¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩) ↔ ¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x)))
9	4, 8	bibi12d 477	. . . 4 ⊢ (⟨w, 0R⟩ = x → ((⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ ¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩)) ↔ (x < ⟨v, 0R⟩ ↔ ¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x))))
10	4	anbi1d 469	. . . . 5 ⊢ (⟨w, 0R⟩ = x → ((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) ↔ (x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩)))
11		breq1 2065	. . . . 5 ⊢ (⟨w, 0R⟩ = x → (⟨w, 0R⟩ < ⟨u, 0R⟩ ↔ x < ⟨u, 0R⟩))
12	10, 11	imbi12d 474	. . . 4 ⊢ (⟨w, 0R⟩ = x → (((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → ⟨w, 0R⟩ < ⟨u, 0R⟩) ↔ ((x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → x < ⟨u, 0R⟩)))
13	9, 12	anbi12d 476	. . 3 ⊢ (⟨w, 0R⟩ = x → (((⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ ¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩)) ∧ ((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → ⟨w, 0R⟩ < ⟨u, 0R⟩)) ↔ ((x < ⟨v, 0R⟩ ↔ ¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x)) ∧ ((x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → x < ⟨u, 0R⟩))))
14		breq2 2066	. . . . 5 ⊢ (⟨v, 0R⟩ = y → (x < ⟨v, 0R⟩ ↔ x < y))
15		cleq2 1110	. . . . . . 7 ⊢ (⟨v, 0R⟩ = y → (x = ⟨v, 0R⟩ ↔ x = y))
16		breq1 2065	. . . . . . 7 ⊢ (⟨v, 0R⟩ = y → (⟨v, 0R⟩ < x ↔ y < x))
17	15, 16	orbi12d 475	. . . . . 6 ⊢ (⟨v, 0R⟩ = y → ((x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x) ↔ (x = y ∨ y < x)))
18	17	negbid 463	. . . . 5 ⊢ (⟨v, 0R⟩ = y → (¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x) ↔ ¬ (x = y ∨ y < x)))
19	14, 18	bibi12d 477	. . . 4 ⊢ (⟨v, 0R⟩ = y → ((x < ⟨v, 0R⟩ ↔ ¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x)) ↔ (x < y ↔ ¬ (x = y ∨ y < x))))
20		breq1 2065	. . . . . 6 ⊢ (⟨v, 0R⟩ = y → (⟨v, 0R⟩ < ⟨u, 0R⟩ ↔ y < ⟨u, 0R⟩))
21	14, 20	anbi12d 476	. . . . 5 ⊢ (⟨v, 0R⟩ = y → ((x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) ↔ (x < y ∧ y < ⟨u, 0R⟩)))
22	21	imbi1d 465	. . . 4 ⊢ (⟨v, 0R⟩ = y → (((x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → x < ⟨u, 0R⟩) ↔ ((x < y ∧ y < ⟨u, 0R⟩) → x < ⟨u, 0R⟩)))
23	19, 22	anbi12d 476	. . 3 ⊢ (⟨v, 0R⟩ = y → (((x < ⟨v, 0R⟩ ↔ ¬ (x = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < x)) ∧ ((x < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → x < ⟨u, 0R⟩)) ↔ ((x < y ↔ ¬ (x = y ∨ y < x)) ∧ ((x < y ∧ y < ⟨u, 0R⟩) → x < ⟨u, 0R⟩))))
24		breq2 2066	. . . . . 6 ⊢ (⟨u, 0R⟩ = z → (y < ⟨u, 0R⟩ ↔ y < z))
25	24	anbi2d 468	. . . . 5 ⊢ (⟨u, 0R⟩ = z → ((x < y ∧ y < ⟨u, 0R⟩) ↔ (x < y ∧ y < z)))
26		breq2 2066	. . . . 5 ⊢ (⟨u, 0R⟩ = z → (x < ⟨u, 0R⟩ ↔ x < z))
27	25, 26	imbi12d 474	. . . 4 ⊢ (⟨u, 0R⟩ = z → (((x < y ∧ y < ⟨u, 0R⟩) → x < ⟨u, 0R⟩) ↔ ((x < y ∧ y < z) → x < z)))
28	27	anbi2d 468	. . 3 ⊢ (⟨u, 0R⟩ = z → (((x < y ↔ ¬ (x = y ∨ y < x)) ∧ ((x < y ∧ y < ⟨u, 0R⟩) → x < ⟨u, 0R⟩)) ↔ ((x < y ↔ ¬ (x = y ∨ y < x)) ∧ ((x < y ∧ y < z) → x < z))))
29		ltsosr 3997	. . . . . . 7 ⊢ <R Or R
30		sotric 2148	. . . . . . 7 ⊢ (( <R Or R ∧ (w ∈ R ∧ v ∈ R)) → (w <R v ↔ ¬ (w = v ∨ v <R w)))
31	29, 30	mpan 518	. . . . . 6 ⊢ ((w ∈ R ∧ v ∈ R) → (w <R v ↔ ¬ (w = v ∨ v <R w)))
32		visset 1350	. . . . . . 7 ⊢ w ∈ V
33		visset 1350	. . . . . . 7 ⊢ v ∈ V
34	32, 33	ltresr 4052	. . . . . 6 ⊢ (⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ w <R v)
35	32	eqresr 4049	. . . . . . . 8 ⊢ (⟨w, 0R⟩ = ⟨v, 0R⟩ ↔ w = v)
36	33, 32	ltresr 4052	. . . . . . . 8 ⊢ (⟨v, 0R⟩ < ⟨w, 0R⟩ ↔ v <R w)
37	35, 36	orbi12i 216	. . . . . . 7 ⊢ ((⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩) ↔ (w = v ∨ v <R w))
38	37	negbii 162	. . . . . 6 ⊢ (¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩) ↔ ¬ (w = v ∨ v <R w))
39	31, 34, 38	3bitr4g 428	. . . . 5 ⊢ ((w ∈ R ∧ v ∈ R) → (⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ ¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩)))
40	39	3adant3 599	. . . 4 ⊢ ((w ∈ R ∧ v ∈ R ∧ u ∈ R) → (⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ ¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩)))
41		ltrelsr 3974	. . . . . 6 ⊢ <R ⊆ (R × R)
42		visset 1350	. . . . . 6 ⊢ u ∈ V
43	32, 29, 41, 33, 42	sotri 2630	. . . . 5 ⊢ ((w <R v ∧ v <R u) → w <R u)
44	33, 42	ltresr 4052	. . . . . 6 ⊢ (⟨v, 0R⟩ < ⟨u, 0R⟩ ↔ v <R u)
45	34, 44	anbi12i 369	. . . . 5 ⊢ ((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) ↔ (w <R v ∧ v <R u))
46	32, 42	ltresr 4052	. . . . 5 ⊢ (⟨w, 0R⟩ < ⟨u, 0R⟩ ↔ w <R u)
47	43, 45, 46	3imtr4 192	. . . 4 ⊢ ((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → ⟨w, 0R⟩ < ⟨u, 0R⟩)
48	40, 47	jctir 241	. . 3 ⊢ ((w ∈ R ∧ v ∈ R ∧ u ∈ R) → ((⟨w, 0R⟩ < ⟨v, 0R⟩ ↔ ¬ (⟨w, 0R⟩ = ⟨v, 0R⟩ ∨ ⟨v, 0R⟩ < ⟨w, 0R⟩)) ∧ ((⟨w, 0R⟩ < ⟨v, 0R⟩ ∧ ⟨v, 0R⟩ < ⟨u, 0R⟩) → ⟨w, 0R⟩ < ⟨u, 0R⟩)))
49	1, 2, 3, 13, 23, 28, 48	3gencl 1367	. 2 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ ∧ z ∈ ℝ) → ((x < y ↔ ¬ (x = y ∨ y < x)) ∧ ((x < y ∧ y < z) → x < z)))
50	49	so 2152	1 ⊢ < Or ℝ

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054   Or wor 2059  Rcnr 3787  0_Rc0r 3788   <_R cltr 3793  ℝcr 4027   < clt 4033

This theorem is referenced by:  axlttri 4083  axlttrn 4084

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-enr 3960  df-nr 3961  df-ltr 3964  df-0r 3965  df-r 4038  df-lt 4041

metamath.org