Theorem zindbi 40684

Description: Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Hypotheses

Ref	Expression
zindbi.1	⊢ (𝑦 ∈ ℤ → (𝜓 ↔ 𝜒))
zindbi.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
zindbi.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒))
zindbi.4	⊢ (𝑥 = 0 → (𝜑 ↔ 𝜃))
zindbi.5	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))

Assertion

Ref	Expression
zindbi	⊢ (𝐴 ∈ ℤ → (𝜃 ↔ 𝜏))

Distinct variable groups:   𝜑,𝑦   𝑥,𝐴,𝑦   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)

Proof of Theorem zindbi

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		c0ex 10900	. . . 4 ⊢ 0 ∈ V
2		zindbi.4	. . . 4 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜃))
3	1, 2	sbcie 3754	. . 3 ⊢ ([0 / 𝑥]𝜑 ↔ 𝜃)
4		0z 12260	. . . . 5 ⊢ 0 ∈ ℤ
5		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 ∈ ℤ ↔ 0 ∈ ℤ))
6		breq1 5073	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑏 ↔ 0 ≤ 𝑏))
7	5, 6	3anbi13d 1436	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) ↔ (0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤ 𝑏)))
8		dfsbcq 3713	. . . . . . . . . 10 ⊢ (𝑦 = 0 → ([𝑦 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))
9	8	bibi1d 343	. . . . . . . . 9 ⊢ (𝑦 = 0 → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)))
10	7, 9	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤ 𝑏) → ([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))))
11		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (𝑏 ∈ ℤ ↔ 𝐴 ∈ ℤ))
12		breq2 5074	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (0 ≤ 𝑏 ↔ 0 ≤ 𝐴))
13	11, 12	3anbi23d 1437	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → ((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤ 𝑏) ↔ (0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)))
14		dfsbcq 3713	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → ([𝑏 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
15	14	bibi2d 342	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)))
16	13, 15	imbi12d 344	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤ 𝑏) → ([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))))
17		dfsbcq 3713	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ([𝑎 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18	17	bibi2d 342	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
19		dfsbcq 3713	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ([𝑎 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))
20	19	bibi2d 342	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)))
21		dfsbcq 3713	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → ([𝑎 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))
22	21	bibi2d 342	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 + 1) → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)))
23		biidd 261	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
24		vex 3426	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
25		zindbi.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
26	24, 25	sbcie 3754	. . . . . . . . . . . . . . 15 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
27		dfsbcq 3713	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))
28	26, 27	bitr3id 284	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (𝜓 ↔ [𝑏 / 𝑥]𝜑))
29		ovex 7288	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 + 1) ∈ V
30		zindbi.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒))
31	29, 30	sbcie 3754	. . . . . . . . . . . . . . 15 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒)
32		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑦 + 1) = (𝑏 + 1))
33	32	sbceq1d 3716	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))
34	31, 33	bitr3id 284	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (𝜒 ↔ [(𝑏 + 1) / 𝑥]𝜑))
35	28, 34	bibi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → ((𝜓 ↔ 𝜒) ↔ ([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)))
36		zindbi.1	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝜓 ↔ 𝜒))
37	35, 36	vtoclga 3503	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℤ → ([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))
38	37	3ad2ant2 1132	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))
39	38	bibi2d 342	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)))
40	39	biimpd 228	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)))
41	18, 20, 22, 20, 23, 40	uzind 12342	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))
42	10, 16, 41	vtocl2g 3500	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)))
43	42	3adant3 1130	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)))
44	43	pm2.43i 52	. . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
45	4, 44	mp3an1 1446	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
46		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ℤ ↔ 𝐴 ∈ ℤ))
47		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ≤ 𝑏 ↔ 𝐴 ≤ 𝑏))
48	46, 47	3anbi13d 1436	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) ↔ (𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏)))
49		dfsbcq 3713	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
50	49	bibi1d 343	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)))
51	48, 50	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) → ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))))
52		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → (𝑏 ∈ ℤ ↔ 0 ∈ ℤ))
53		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → (𝐴 ≤ 𝑏 ↔ 𝐴 ≤ 0))
54	52, 53	3anbi23d 1437	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) ↔ (𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0)))
55		dfsbcq 3713	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ([𝑏 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))
56	55	bibi2d 342	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)))
57	54, 56	imbi12d 344	. . . . . . . . 9 ⊢ (𝑏 = 0 → (((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) → ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))))
58	51, 57, 41	vtocl2g 3500	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)))
59	58	3adant3 1130	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)))
60	59	pm2.43i 52	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))
61	4, 60	mp3an2 1447	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))
62	61	bicomd 222	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≤ 0) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
63		0re 10908	. . . . 5 ⊢ 0 ∈ ℝ
64		zre 12253	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
65		letric 11005	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
66	63, 64, 65	sylancr 586	. . . 4 ⊢ (𝐴 ∈ ℤ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
67	45, 62, 66	mpjaodan 955	. . 3 ⊢ (𝐴 ∈ ℤ → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
68	3, 67	bitr3id 284	. 2 ⊢ (𝐴 ∈ ℤ → (𝜃 ↔ [𝐴 / 𝑥]𝜑))
69		zindbi.5	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
70	69	sbcieg 3751	. 2 ⊢ (𝐴 ∈ ℤ → ([𝐴 / 𝑥]𝜑 ↔ 𝜏))
71	68, 70	bitrd 278	1 ⊢ (𝐴 ∈ ℤ → (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  [wsbc 3711   class class class wbr 5070  (class class class)co 7255  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   ≤ cle 10941  ℤcz 12249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250

This theorem is referenced by:  jm2.25  40737