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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 42248 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 1915 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
6 | 4, 5 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
7 | climinff.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
8 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
9 | 7, 8 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
10 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
11 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 7, 11 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | 9, 10, 12 | nfbr 5077 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗) |
14 | 6, 13 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
15 | eleq1w 2872 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 631 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fvoveq1 7158 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) | |
18 | fveq2 6645 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
19 | 17, 18 | breq12d 5043 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))) |
20 | 16, 19 | imbi12d 348 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)))) |
21 | climinff.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) | |
22 | 14, 20, 21 | chvarfv 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
23 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑘ℝ | |
24 | 5 | nfci 2939 | . . . . . 6 ⊢ Ⅎ𝑘𝑍 |
25 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑘𝑥 | |
26 | 25, 10, 12 | nfbr 5077 | . . . . . 6 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑗) |
27 | 24, 26 | nfralw 3189 | . . . . 5 ⊢ Ⅎ𝑘∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) |
28 | 23, 27 | nfrex 3268 | . . . 4 ⊢ Ⅎ𝑘∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) |
29 | 4, 28 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘(𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
30 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑘) | |
31 | 18 | breq2d 5042 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑥 ≤ (𝐹‘𝑘) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
32 | 30, 26, 31 | cbvralw 3387 | . . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝑘 = 𝑗 → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
34 | 33 | rexbidv 3256 | . . . 4 ⊢ (𝑘 = 𝑗 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
35 | 34 | imbi2d 344 | . . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ↔ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) |
36 | climinff.7 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) | |
37 | 29, 35, 36 | chvarfv 2240 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
38 | 1, 2, 3, 22, 37 | climinf 42248 | 1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: (None) |
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