Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > climinff | Structured version Visualization version GIF version |
Description: A version of climinf 41907 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 2977 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
9 | 7, 8 | nffv 6680 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
10 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
11 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 7, 11 | nffv 6680 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | 9, 10, 12 | nfbr 5113 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗) |
14 | 6, 13 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
15 | eleq1w 2895 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fvoveq1 7179 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) | |
18 | fveq2 6670 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
19 | 17, 18 | breq12d 5079 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))) |
20 | 16, 19 | imbi12d 347 | . . 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 2242 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
23 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘ℝ | |
24 | 5 | nfci 2964 | . . . . . 6 ⊢ Ⅎ𝑘𝑍 |
25 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘𝑥 | |
26 | 25, 10, 12 | nfbr 5113 | . . . . . 6 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑗) |
27 | 24, 26 | nfralw 3225 | . . . . 5 ⊢ Ⅎ𝑘∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) |
28 | 23, 27 | nfrex 3309 | . . . 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 5078 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑥 ≤ (𝐹‘𝑘) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
32 | 30, 26, 31 | cbvralw 3441 | . . . . . 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 3297 | . . . 4 ⊢ (𝑘 = 𝑗 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
35 | 34 | imbi2d 343 | . . 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 2242 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
38 | 1, 2, 3, 22, 37 | climinf 41907 | 1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 infcinf 8905 ℝcr 10536 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |