Theorem fzind 11351

Description: Induction on the integers from 𝑀 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)

Hypotheses

Ref	Expression
fzind.1	⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))
fzind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
fzind.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
fzind.4	⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))
fzind.5	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓)
fzind.6	⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))

Assertion

Ref	Expression
fzind	⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)

Distinct variable groups:   𝑥,𝐾   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥   𝜏,𝑥   𝜃,𝑥

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

Proof of Theorem fzind

Step	Hyp	Ref	Expression
1		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 ≤ 𝑁 ↔ 𝑀 ≤ 𝑁))
2	1	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))
3		fzind.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))
4	2, 3	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓)))
5		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑁 ↔ 𝑦 ≤ 𝑁))
6	5	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁)))
7		fzind.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
8	6, 7	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒)))
9		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑁 ↔ (𝑦 + 1) ≤ 𝑁))
10	9	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁)))
11		fzind.3	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
12	10, 11	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃)))
13		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐾 → (𝑥 ≤ 𝑁 ↔ 𝐾 ≤ 𝑁))
14	13	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑥 = 𝐾 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁)))
15		fzind.4	. . . . . . . . . 10 ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))
16	14, 15	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 𝐾 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁) → 𝜏)))
17		fzind.5	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓)
18	17	3expib 1260	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓))
19		zre 11258	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
20		zre 11258	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
21		p1le 10745	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑦 + 1) ≤ 𝑁) → 𝑦 ≤ 𝑁)
22	21	3expia 1259	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑁 → 𝑦 ≤ 𝑁))
23	19, 20, 22	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑦 + 1) ≤ 𝑁 → 𝑦 ≤ 𝑁))
24	23	imdistanda 725	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → (𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁)))
25	24	imim1d 80	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒)))
26	25	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒)))
27		zltp1le 11304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁))
28	27	adantlr 747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑁 ∈ ℤ) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁))
29	28	expcom 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁)))
30	29	pm5.32d 669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁)))
31	30	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁)))
32		fzind.6	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
33	32	expcom 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝜒 → 𝜃)))
34	33	3expa 1257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝜒 → 𝜃)))
35	34	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃)))
36	31, 35	sylbird 249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃)))
37	36	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃))))
38	37	com23 84	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝑁 ∈ ℤ → (𝜒 → 𝜃))))
39	38	expd 451	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑦 + 1) ≤ 𝑁 → (𝑁 ∈ ℤ → (𝜒 → 𝜃)))))
40	39	3impib 1254	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑦 + 1) ≤ 𝑁 → (𝑁 ∈ ℤ → (𝜒 → 𝜃))))
41	40	com23 84	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (𝑁 ∈ ℤ → ((𝑦 + 1) ≤ 𝑁 → (𝜒 → 𝜃))))
42	41	impd 446	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃)))
43	42	a2d 29	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃)))
44	26, 43	syld 46	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃)))
45	4, 8, 12, 16, 18, 44	uzind 11345	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → ((𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁) → 𝜏))
46	45	expcomd 453	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏)))
47	46	3expb 1258	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾)) → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏)))
48	47	expcom 450	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝑀 ∈ ℤ → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏))))
49	48	com23 84	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝐾 ≤ 𝑁 → (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → 𝜏))))
50	49	3impia 1253	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → 𝜏)))
51	50	impd 446	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝜏))
52	51	impcom 445	1 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  (class class class)co 6549  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℤcz 11254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255

This theorem is referenced by:  fnn0ind  11352  fzind2  12448  ssinc  38292  ssdec  38293