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Theorem nnsub 11673

Description: Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

**Assertion**
Ref	Expression
nnsub	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ))

Proof of Theorem nnsub Dummy variables 𝑧 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5061	. . . . . 6 ⊢ (𝑥 = 1 → (𝑧 < 𝑥 ↔ 𝑧 < 1))
2		oveq1 7155	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 − 𝑧) = (1 − 𝑧))
3	2	eleq1d 2895	. . . . . 6 ⊢ (𝑥 = 1 → ((𝑥 − 𝑧) ∈ ℕ ↔ (1 − 𝑧) ∈ ℕ))
4	1, 3	imbi12d 347	. . . . 5 ⊢ (𝑥 = 1 → ((𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ (𝑧 < 1 → (1 − 𝑧) ∈ ℕ)))
5	4	ralbidv 3195	. . . 4 ⊢ (𝑥 = 1 → (∀𝑧 ∈ ℕ (𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ ∀𝑧 ∈ ℕ (𝑧 < 1 → (1 − 𝑧) ∈ ℕ)))
6		breq2 5061	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 < 𝑥 ↔ 𝑧 < 𝑦))
7		oveq1 7155	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑧) = (𝑦 − 𝑧))
8	7	eleq1d 2895	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑧) ∈ ℕ ↔ (𝑦 − 𝑧) ∈ ℕ))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ (𝑧 < 𝑦 → (𝑦 − 𝑧) ∈ ℕ)))
10	9	ralbidv 3195	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ (𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ ∀𝑧 ∈ ℕ (𝑧 < 𝑦 → (𝑦 − 𝑧) ∈ ℕ)))
11		breq2 5061	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑧 < 𝑥 ↔ 𝑧 < (𝑦 + 1)))
12		oveq1 7155	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 𝑧) = ((𝑦 + 1) − 𝑧))
13	12	eleq1d 2895	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 𝑧) ∈ ℕ ↔ ((𝑦 + 1) − 𝑧) ∈ ℕ))
14	11, 13	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
15	14	ralbidv 3195	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑧 ∈ ℕ (𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
16		breq2 5061	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑧 < 𝑥 ↔ 𝑧 < 𝐵))
17		oveq1 7155	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 − 𝑧) = (𝐵 − 𝑧))
18	17	eleq1d 2895	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 − 𝑧) ∈ ℕ ↔ (𝐵 − 𝑧) ∈ ℕ))
19	16, 18	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ (𝑧 < 𝐵 → (𝐵 − 𝑧) ∈ ℕ)))
20	19	ralbidv 3195	. . . 4 ⊢ (𝑥 = 𝐵 → (∀𝑧 ∈ ℕ (𝑧 < 𝑥 → (𝑥 − 𝑧) ∈ ℕ) ↔ ∀𝑧 ∈ ℕ (𝑧 < 𝐵 → (𝐵 − 𝑧) ∈ ℕ)))
21		nnnlt1 11661	. . . . . 6 ⊢ (𝑧 ∈ ℕ → ¬ 𝑧 < 1)
22	21	pm2.21d 121	. . . . 5 ⊢ (𝑧 ∈ ℕ → (𝑧 < 1 → (1 − 𝑧) ∈ ℕ))
23	22	rgen 3146	. . . 4 ⊢ ∀𝑧 ∈ ℕ (𝑧 < 1 → (1 − 𝑧) ∈ ℕ)
24		breq1 5060	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 < 𝑦 ↔ 𝑥 < 𝑦))
25		oveq2 7156	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 − 𝑧) = (𝑦 − 𝑥))
26	25	eleq1d 2895	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 − 𝑧) ∈ ℕ ↔ (𝑦 − 𝑥) ∈ ℕ))
27	24, 26	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 < 𝑦 → (𝑦 − 𝑧) ∈ ℕ) ↔ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ)))
28	27	cbvralvw 3448	. . . . 5 ⊢ (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → (𝑦 − 𝑧) ∈ ℕ) ↔ ∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ))
29		nncn 11638	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
30	29	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑦 ∈ ℂ)
31		ax-1cn 10587	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
32		pncan 10884	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦)
33	30, 31, 32	sylancl 588	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑦 + 1) − 1) = 𝑦)
34		simpl 485	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑦 ∈ ℕ)
35	33, 34	eqeltrd 2911	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑦 + 1) − 1) ∈ ℕ)
36		oveq2 7156	. . . . . . . . . 10 ⊢ (𝑧 = 1 → ((𝑦 + 1) − 𝑧) = ((𝑦 + 1) − 1))
37	36	eleq1d 2895	. . . . . . . . 9 ⊢ (𝑧 = 1 → (((𝑦 + 1) − 𝑧) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ))
38	35, 37	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧 = 1 → ((𝑦 + 1) − 𝑧) ∈ ℕ))
39	38	2a1dd 51	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧 = 1 → (∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) → (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ))))
40		breq1 5060	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 − 1) → (𝑥 < 𝑦 ↔ (𝑧 − 1) < 𝑦))
41		oveq2 7156	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 − 1) → (𝑦 − 𝑥) = (𝑦 − (𝑧 − 1)))
42	41	eleq1d 2895	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 − 1) → ((𝑦 − 𝑥) ∈ ℕ ↔ (𝑦 − (𝑧 − 1)) ∈ ℕ))
43	40, 42	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 − 1) → ((𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) ↔ ((𝑧 − 1) < 𝑦 → (𝑦 − (𝑧 − 1)) ∈ ℕ)))
44	43	rspcv 3616	. . . . . . . 8 ⊢ ((𝑧 − 1) ∈ ℕ → (∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) → ((𝑧 − 1) < 𝑦 → (𝑦 − (𝑧 − 1)) ∈ ℕ)))
45		nnre 11637	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
46		nnre 11637	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
47		1re 10633	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
48		ltsubadd 11102	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 − 1) < 𝑦 ↔ 𝑧 < (𝑦 + 1)))
49	47, 48	mp3an2 1443	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 − 1) < 𝑦 ↔ 𝑧 < (𝑦 + 1)))
50	45, 46, 49	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 − 1) < 𝑦 ↔ 𝑧 < (𝑦 + 1)))
51		nncn 11638	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ)
52		subsub3 10910	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑦 − (𝑧 − 1)) = ((𝑦 + 1) − 𝑧))
53	31, 52	mp3an3 1444	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 − (𝑧 − 1)) = ((𝑦 + 1) − 𝑧))
54	29, 51, 53	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑦 − (𝑧 − 1)) = ((𝑦 + 1) − 𝑧))
55	54	eleq1d 2895	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑦 − (𝑧 − 1)) ∈ ℕ ↔ ((𝑦 + 1) − 𝑧) ∈ ℕ))
56	50, 55	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (((𝑧 − 1) < 𝑦 → (𝑦 − (𝑧 − 1)) ∈ ℕ) ↔ (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
57	56	biimpd 231	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (((𝑧 − 1) < 𝑦 → (𝑦 − (𝑧 − 1)) ∈ ℕ) → (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
58	44, 57	syl9r 78	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 − 1) ∈ ℕ → (∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) → (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ))))
59		nn1m1nn 11650	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 = 1 ∨ (𝑧 − 1) ∈ ℕ))
60	59	adantl 484	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧 = 1 ∨ (𝑧 − 1) ∈ ℕ))
61	39, 58, 60	mpjaod 856	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) → (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
62	61	ralrimdva 3187	. . . . 5 ⊢ (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ (𝑥 < 𝑦 → (𝑦 − 𝑥) ∈ ℕ) → ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
63	28, 62	syl5bi 244	. . . 4 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → (𝑦 − 𝑧) ∈ ℕ) → ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ((𝑦 + 1) − 𝑧) ∈ ℕ)))
64	5, 10, 15, 20, 23, 63	nnind 11648	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑧 ∈ ℕ (𝑧 < 𝐵 → (𝐵 − 𝑧) ∈ ℕ))
65		breq1 5060	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 < 𝐵 ↔ 𝐴 < 𝐵))
66		oveq2 7156	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝐵 − 𝑧) = (𝐵 − 𝐴))
67	66	eleq1d 2895	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝐵 − 𝑧) ∈ ℕ ↔ (𝐵 − 𝐴) ∈ ℕ))
68	65, 67	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 < 𝐵 → (𝐵 − 𝑧) ∈ ℕ) ↔ (𝐴 < 𝐵 → (𝐵 − 𝐴) ∈ ℕ)))
69	68	rspcva 3619	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ ∀𝑧 ∈ ℕ (𝑧 < 𝐵 → (𝐵 − 𝑧) ∈ ℕ)) → (𝐴 < 𝐵 → (𝐵 − 𝐴) ∈ ℕ))
70	64, 69	sylan2 594	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → (𝐵 − 𝐴) ∈ ℕ))
71		nngt0 11660	. . 3 ⊢ ((𝐵 − 𝐴) ∈ ℕ → 0 < (𝐵 − 𝐴))
72		nnre 11637	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
73		nnre 11637	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
74		posdif 11125	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
75	72, 73, 74	syl2an 597	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
76	71, 75	syl5ibr 248	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 − 𝐴) ∈ ℕ → 𝐴 < 𝐵))
77	70, 76	impbid 214	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1531 ∈ wcel 2108 ∀wral 3136 class class class wbr 5057 (class class class)co 7148 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 < clt 10667 − cmin 10862 ℕcn 11630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631

This theorem is referenced by: nnsubi 11674 nn0sub 11939 uz3m2nn 12283 faclbnd4lem4 13648 pythagtriplem13 16156 vdwlem12 16320 perfectlem1 25797 crctcshwlkn0lem6 27585 crctcshwlkn0lem7 27586 bcprod 32963 nndivsub 33798 perfectALTVlem1 43877

W3C validator