Theorem exbtwnzlemshrink 10355

Description: Lemma for exbtwnzlemex 10356. Shrinking the range around 𝐴. (Contributed by Jim Kingdon, 10-May-2022.)

Hypotheses

Ref	Expression
exbtwnzlemshrink.j	⊢ (𝜑 → 𝐽 ∈ ℕ)
exbtwnzlemshrink.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
exbtwnzlemshrink.tri	⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))

Assertion

Ref	Expression
exbtwnzlemshrink	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Distinct variable groups:   𝐴,𝑚,𝑛   𝑥,𝐴,𝑚   𝑚,𝐽   𝜑,𝑚,𝑛

Allowed substitution hints:   𝜑(𝑥)   𝐽(𝑥,𝑛)

Proof of Theorem exbtwnzlemshrink

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exbtwnzlemshrink.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ)
2	1	adantr 276	. 2 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → 𝐽 ∈ ℕ)
3		oveq2 5933	. . . . . . . 8 ⊢ (𝑤 = 1 → (𝑚 + 𝑤) = (𝑚 + 1))
4	3	breq2d 4046	. . . . . . 7 ⊢ (𝑤 = 1 → (𝐴 < (𝑚 + 𝑤) ↔ 𝐴 < (𝑚 + 1)))
5	4	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 1 → ((𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1))))
6	5	rexbidv 2498	. . . . 5 ⊢ (𝑤 = 1 → (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1))))
7	6	anbi2d 464	. . . 4 ⊢ (𝑤 = 1 → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) ↔ (𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1)))))
8	7	imbi1d 231	. . 3 ⊢ (𝑤 = 1 → (((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) ↔ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))))
9		oveq2 5933	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (𝑚 + 𝑤) = (𝑚 + 𝑘))
10	9	breq2d 4046	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝐴 < (𝑚 + 𝑤) ↔ 𝐴 < (𝑚 + 𝑘)))
11	10	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑘 → ((𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘))))
12	11	rexbidv 2498	. . . . 5 ⊢ (𝑤 = 𝑘 → (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘))))
13	12	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) ↔ (𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘)))))
14	13	imbi1d 231	. . 3 ⊢ (𝑤 = 𝑘 → (((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) ↔ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))))
15		oveq2 5933	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (𝑚 + 𝑤) = (𝑚 + (𝑘 + 1)))
16	15	breq2d 4046	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝐴 < (𝑚 + 𝑤) ↔ 𝐴 < (𝑚 + (𝑘 + 1))))
17	16	anbi2d 464	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → ((𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))))
18	17	rexbidv 2498	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))))
19	18	anbi2d 464	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) ↔ (𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1))))))
20	19	imbi1d 231	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) ↔ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))))
21		oveq2 5933	. . . . . . . 8 ⊢ (𝑤 = 𝐽 → (𝑚 + 𝑤) = (𝑚 + 𝐽))
22	21	breq2d 4046	. . . . . . 7 ⊢ (𝑤 = 𝐽 → (𝐴 < (𝑚 + 𝑤) ↔ 𝐴 < (𝑚 + 𝐽)))
23	22	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝐽 → ((𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))))
24	23	rexbidv 2498	. . . . 5 ⊢ (𝑤 = 𝐽 → (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤)) ↔ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))))
25	24	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝐽 → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) ↔ (𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽)))))
26	25	imbi1d 231	. . 3 ⊢ (𝑤 = 𝐽 → (((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑤))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) ↔ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))))
27		breq1 4037	. . . . . . 7 ⊢ (𝑚 = 𝑥 → (𝑚 ≤ 𝐴 ↔ 𝑥 ≤ 𝐴))
28		oveq1 5932	. . . . . . . 8 ⊢ (𝑚 = 𝑥 → (𝑚 + 1) = (𝑥 + 1))
29	28	breq2d 4046	. . . . . . 7 ⊢ (𝑚 = 𝑥 → (𝐴 < (𝑚 + 1) ↔ 𝐴 < (𝑥 + 1)))
30	27, 29	anbi12d 473	. . . . . 6 ⊢ (𝑚 = 𝑥 → ((𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1)) ↔ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
31	30	cbvrexv 2730	. . . . 5 ⊢ (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1)) ↔ ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
32	31	biimpi 120	. . . 4 ⊢ (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1)) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
33	32	adantl 277	. . 3 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 1))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
34		simpl 109	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝜑) → 𝑘 ∈ ℕ)
35		exbtwnzlemshrink.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
36	35	adantl 277	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝜑) → 𝐴 ∈ ℝ)
37		exbtwnzlemshrink.tri	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
38	37	adantll 476	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
39	34, 36, 38	exbtwnzlemstep 10354	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘)))
40	39	ex 115	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ 𝜑) → (∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘))))
41	40	imdistanda 448	. . . 4 ⊢ (𝑘 ∈ ℕ → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))) → (𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘)))))
42	41	imim1d 75	. . 3 ⊢ (𝑘 ∈ ℕ → (((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝑘))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑘 + 1)))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))))
43	8, 14, 20, 26, 33, 42	nnind 9023	. 2 ⊢ (𝐽 ∈ ℕ → ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
44	2, 43	mpcom 36	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   class class class wbr 4034  (class class class)co 5925  ℝcr 7895  1c1 7897   + caddc 7899   < clt 8078   ≤ cle 8079  ℕcn 9007  ℤcz 9343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344

This theorem is referenced by:  exbtwnzlemex  10356