Theorem nn0ltp1let 4556

Description: Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-02.)

Assertion

Ref	Expression
nn0ltp1let	⊢ ((A ∈ ℕ0 ∧ B ∈ ℕ0) → (A < B ↔ (A + 1) ≤ B))

Proof of Theorem nn0ltp1let

Step	Hyp	Ref	Expression
1		nnltp1let 4449	. . 3 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A < B ↔ (A + 1) ≤ B))
2		breq1 2065	. . . . 5 ⊢ (A = 0 → (A < B ↔ 0 < B))
3		nngt0t 4441	. . . . . . 7 ⊢ (B ∈ ℕ → 0 < B)
4		nnge1t 4439	. . . . . . . 8 ⊢ (B ∈ ℕ → 1 ≤ B)
5		1cn 4101	. . . . . . . . 9 ⊢ 1 ∈ ℂ
6	5	addid2 4113	. . . . . . . 8 ⊢ (0 + 1) = 1
7	4, 6	syl5eqbr 2089	. . . . . . 7 ⊢ (B ∈ ℕ → (0 + 1) ≤ B)
8	3, 7	2th 540	. . . . . 6 ⊢ ((B ∈ ℕ → 0 < B) ↔ (B ∈ ℕ → (0 + 1) ≤ B))
9	8	pm5.74ri 445	. . . . 5 ⊢ (B ∈ ℕ → (0 < B ↔ (0 + 1) ≤ B))
10	2, 9	sylan9bb 418	. . . 4 ⊢ ((A = 0 ∧ B ∈ ℕ) → (A < B ↔ (0 + 1) ≤ B))
11		opreq1 3006	. . . . . 6 ⊢ (A = 0 → (A + 1) = (0 + 1))
12	11	breq1d 2071	. . . . 5 ⊢ (A = 0 → ((A + 1) ≤ B ↔ (0 + 1) ≤ B))
13	12	adantr 306	. . . 4 ⊢ ((A = 0 ∧ B ∈ ℕ) → ((A + 1) ≤ B ↔ (0 + 1) ≤ B))
14	10, 13	bitr4d 409	. . 3 ⊢ ((A = 0 ∧ B ∈ ℕ) → (A < B ↔ (A + 1) ≤ B))
15		breq2 2066	. . . . 5 ⊢ (B = 0 → (A < B ↔ A < 0))
16		pm5.21 502	. . . . . 6 ⊢ ((¬ A < 0 ∧ ¬ (A + 1) ≤ 0) → (A < 0 ↔ (A + 1) ≤ 0))
17		nngt0t 4441	. . . . . . 7 ⊢ (A ∈ ℕ → 0 < A)
18		nnret 4427	. . . . . . . 8 ⊢ (A ∈ ℕ → A ∈ ℝ)
19		ax0re 4063	. . . . . . . . 9 ⊢ 0 ∈ ℝ
20		ltnsymt 4294	. . . . . . . . 9 ⊢ ((A ∈ ℝ ∧ 0 ∈ ℝ) → (A < 0 → ¬ 0 < A))
21	19, 20	mpan2 519	. . . . . . . 8 ⊢ (A ∈ ℝ → (A < 0 → ¬ 0 < A))
22	18, 21	syl 12	. . . . . . 7 ⊢ (A ∈ ℕ → (A < 0 → ¬ 0 < A))
23	17, 22	mt2d 98	. . . . . 6 ⊢ (A ∈ ℕ → ¬ A < 0)
24		axlttrn 4084	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ A ∈ ℝ ∧ (A + 1) ∈ ℝ) → ((0 < A ∧ A < (A + 1)) → 0 < (A + 1)))
25	19, 24	mp3an1 639	. . . . . . . 8 ⊢ ((A ∈ ℝ ∧ (A + 1) ∈ ℝ) → ((0 < A ∧ A < (A + 1)) → 0 < (A + 1)))
26		ax1re 4064	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
27		axaddrcl 4067	. . . . . . . . . . 11 ⊢ ((A ∈ ℝ ∧ 1 ∈ ℝ) → (A + 1) ∈ ℝ)
28	26, 27	mpan2 519	. . . . . . . . . 10 ⊢ (A ∈ ℝ → (A + 1) ∈ ℝ)
29	18, 28	syl 12	. . . . . . . . 9 ⊢ (A ∈ ℕ → (A + 1) ∈ ℝ)
30	18, 29	jca 236	. . . . . . . 8 ⊢ (A ∈ ℕ → (A ∈ ℝ ∧ (A + 1) ∈ ℝ))
31		ltplus1t 4383	. . . . . . . . . 10 ⊢ (A ∈ ℝ → A < (A + 1))
32	18, 31	syl 12	. . . . . . . . 9 ⊢ (A ∈ ℕ → A < (A + 1))
33	17, 32	jca 236	. . . . . . . 8 ⊢ (A ∈ ℕ → (0 < A ∧ A < (A + 1)))
34	25, 30, 33	sylc 62	. . . . . . 7 ⊢ (A ∈ ℕ → 0 < (A + 1))
35		leltt 4278	. . . . . . . . . 10 ⊢ (((A + 1) ∈ ℝ ∧ 0 ∈ ℝ) → ((A + 1) ≤ 0 ↔ ¬ 0 < (A + 1)))
36	19, 35	mpan2 519	. . . . . . . . 9 ⊢ ((A + 1) ∈ ℝ → ((A + 1) ≤ 0 ↔ ¬ 0 < (A + 1)))
37	18, 28, 36	3syl 21	. . . . . . . 8 ⊢ (A ∈ ℕ → ((A + 1) ≤ 0 ↔ ¬ 0 < (A + 1)))
38	37	bicon2d 404	. . . . . . 7 ⊢ (A ∈ ℕ → (0 < (A + 1) ↔ ¬ (A + 1) ≤ 0))
39	34, 38	mpbid 170	. . . . . 6 ⊢ (A ∈ ℕ → ¬ (A + 1) ≤ 0)
40	16, 23, 39	sylanc 361	. . . . 5 ⊢ (A ∈ ℕ → (A < 0 ↔ (A + 1) ≤ 0))
41	15, 40	sylan9bbr 419	. . . 4 ⊢ ((A ∈ ℕ ∧ B = 0) → (A < B ↔ (A + 1) ≤ 0))
42		breq2 2066	. . . . 5 ⊢ (B = 0 → ((A + 1) ≤ B ↔ (A + 1) ≤ 0))
43	42	adantl 305	. . . 4 ⊢ ((A ∈ ℕ ∧ B = 0) → ((A + 1) ≤ B ↔ (A + 1) ≤ 0))
44	41, 43	bitr4d 409	. . 3 ⊢ ((A ∈ ℕ ∧ B = 0) → (A < B ↔ (A + 1) ≤ B))
45		breq1 2065	. . . . . . 7 ⊢ (A = 0 → (A < 0 ↔ 0 < 0))
46	19	ltnr 4338	. . . . . . . 8 ⊢ ¬ 0 < 0
47	19	ltplus1 4384	. . . . . . . . 9 ⊢ 0 < (0 + 1)
48	19, 26	readdcl 4118	. . . . . . . . . . 11 ⊢ (0 + 1) ∈ ℝ
49	48, 19	lelt 4301	. . . . . . . . . 10 ⊢ ((0 + 1) ≤ 0 ↔ ¬ 0 < (0 + 1))
50	49	bicon2i 194	. . . . . . . . 9 ⊢ (0 < (0 + 1) ↔ ¬ (0 + 1) ≤ 0)
51	47, 50	mpbi 164	. . . . . . . 8 ⊢ ¬ (0 + 1) ≤ 0
52		pm5.21 502	. . . . . . . 8 ⊢ ((¬ 0 < 0 ∧ ¬ (0 + 1) ≤ 0) → (0 < 0 ↔ (0 + 1) ≤ 0))
53	46, 51, 52	mp2an 520	. . . . . . 7 ⊢ (0 < 0 ↔ (0 + 1) ≤ 0)
54	45, 53	syl6bb 414	. . . . . 6 ⊢ (A = 0 → (A < 0 ↔ (0 + 1) ≤ 0))
55	11	breq1d 2071	. . . . . 6 ⊢ (A = 0 → ((A + 1) ≤ 0 ↔ (0 + 1) ≤ 0))
56	54, 55	bitr4d 409	. . . . 5 ⊢ (A = 0 → (A < 0 ↔ (A + 1) ≤ 0))
57	15, 56	sylan9bbr 419	. . . 4 ⊢ ((A = 0 ∧ B = 0) → (A < B ↔ (A + 1) ≤ 0))
58	42	adantl 305	. . . 4 ⊢ ((A = 0 ∧ B = 0) → ((A + 1) ≤ B ↔ (A + 1) ≤ 0))
59	57, 58	bitr4d 409	. . 3 ⊢ ((A = 0 ∧ B = 0) → (A < B ↔ (A + 1) ≤ B))
60	1, 14, 44, 59	ccase 562	. 2 ⊢ (((A ∈ ℕ ∨ A = 0) ∧ (B ∈ ℕ ∨ B = 0)) → (A < B ↔ (A + 1) ≤ B))
61		elnn0 4536	. 2 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))
62		elnn0 4536	. 2 ⊢ (B ∈ ℕ0 ↔ (B ∈ ℕ ∨ B = 0))
63	60, 61, 62	syl2anb 350	1 ⊢ ((A ∈ ℕ0 ∧ B ∈ ℕ0) → (A < B ↔ (A + 1) ≤ B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  (class class class)co 3001  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031   < clt 4033   ≤ cle 4092  ℕcn 4093  ℕ₀cn0 4094

This theorem is referenced by:  nn0leltp1t 4557  nn0ltlem1 4558  zltp1let 4597  nn0opthlem1 4722

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-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535

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