Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climinff | Structured version Visualization version GIF version |
Description: A version of climinf 39838 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
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)) |
Ref | Expression |
---|---|
climinff | ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) |
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 1843 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
6 | 4, 5 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
7 | climinff.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
8 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
9 | 7, 8 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
10 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
11 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 7, 11 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | 9, 10, 12 | nfbr 4699 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗) |
14 | 6, 13 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
15 | eleq1 2689 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 740 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | oveq1 6657 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1)) | |
18 | 17 | fveq2d 6195 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) |
19 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
20 | 18, 19 | breq12d 4666 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))) |
21 | 16, 20 | imbi12d 334 | . . 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 2262 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
24 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘ℝ | |
25 | 5 | nfci 2754 | . . . . . 6 ⊢ Ⅎ𝑘𝑍 |
26 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘𝑥 | |
27 | 26, 10, 12 | nfbr 4699 | . . . . . 6 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑗) |
28 | 25, 27 | nfral 2945 | . . . . 5 ⊢ Ⅎ𝑘∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) |
29 | 24, 28 | nfrex 3007 | . . . 4 ⊢ Ⅎ𝑘∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) |
30 | 4, 29 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘(𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
31 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑘) | |
32 | 19 | breq2d 4665 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑥 ≤ (𝐹‘𝑘) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
33 | 31, 27, 32 | cbvral 3167 | . . . . . 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 3052 | . . . 4 ⊢ (𝑘 = 𝑗 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
36 | 35 | imbi2d 330 | . . 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 2262 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
39 | 1, 2, 3, 23, 38 | climinf 39838 | 1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 infcinf 8347 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 ℤcz 11377 ℤ≥cuz 11687 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 |
This theorem is referenced by: (None) |
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