Theorem climinff 38678

Description: A version of climinf 38673 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)

Hypotheses

Ref	Expression
climinff.1	⊢ Ⅎ𝑘𝜑
climinff.2	⊢ Ⅎ𝑘𝐹
climinff.3	⊢ 𝑍 = (ℤ≥‘𝑀)
climinff.4	⊢ (𝜑 → 𝑀 ∈ ℤ)
climinff.5	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
climinff.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
climinff.7	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))

Assertion

Ref	Expression
climinff	⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))

Distinct variable groups:   𝑥,𝑘   𝑥,𝐹   𝑘,𝑍,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑘)   𝐹(𝑘)   𝑀(𝑥,𝑘)

Proof of Theorem climinff

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climinff.3	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climinff.4	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climinff.5	. 2 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4		climinff.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		nfv 1830	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
6	4, 5	nfan 1816	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
7		climinff.2	. . . . . 6 ⊢ Ⅎ𝑘𝐹
8		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1)
9	7, 8	nffv 6110	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1))
10		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑘 ≤
11		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑘𝑗
12	7, 11	nffv 6110	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
13	9, 10, 12	nfbr 4629	. . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)
14	6, 13	nfim 1813	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))
15		eleq1 2676	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 736	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		oveq1 6556	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1))
18	17	fveq2d 6107	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1)))
19		fveq2 6103	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
20	18, 19	breq12d 4596	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)))
21	16, 20	imbi12d 333	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))))
22		climinff.6	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
23	14, 21, 22	chvar 2250	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))
24		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑘ℝ
25	5	nfci 2741	. . . . . 6 ⊢ Ⅎ𝑘𝑍
26		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
27	26, 10, 12	nfbr 4629	. . . . . 6 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑗)
28	25, 27	nfral 2929	. . . . 5 ⊢ Ⅎ𝑘∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)
29	24, 28	nfrex 2990	. . . 4 ⊢ Ⅎ𝑘∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)
30	4, 29	nfim 1813	. . 3 ⊢ Ⅎ𝑘(𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
31		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑘)
32	19	breq2d 4595	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑥 ≤ (𝐹‘𝑘) ↔ 𝑥 ≤ (𝐹‘𝑗)))
33	31, 27, 32	cbvral 3143	. . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
34	33	a1i 11	. . . . 5 ⊢ (𝑘 = 𝑗 → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
35	34	rexbidv 3034	. . . 4 ⊢ (𝑘 = 𝑗 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
36	35	imbi2d 329	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ↔ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
37		climinff.7	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
38	30, 36, 37	chvar 2250	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
39	1, 2, 3, 23, 38	climinf 38673	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  infcinf 8230  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℤcz 11254  ℤ_≥cuz 11563   ⇝ cli 14063

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  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  ax-pre-sup 9893

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-rmo 2904  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-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067

This theorem is referenced by: (None)