![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fltltc | Structured version Visualization version GIF version |
Description: (𝐶↑𝑁) is the largest term and therefore 𝐵 < 𝐶. (Contributed by Steven Nguyen, 22-Aug-2023.) |
Ref | Expression |
---|---|
fltne.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
fltne.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
fltne.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
fltne.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) |
fltne.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltltc | ⊢ (𝜑 → 𝐵 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltne.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nncnd 11641 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | fltne.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
4 | eluzge3nn 12278 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11943 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | 2, 6 | expcld 13506 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
8 | fltne.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | 8 | nncnd 11641 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | 9, 6 | expcld 13506 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℂ) |
11 | fltne.1 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
12 | 7, 10, 11 | mvlladdd 11040 | . . 3 ⊢ (𝜑 → (𝐵↑𝑁) = ((𝐶↑𝑁) − (𝐴↑𝑁))) |
13 | fltne.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
14 | 13 | nnred 11640 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
15 | 14, 6 | reexpcld 13523 | . . . 4 ⊢ (𝜑 → (𝐶↑𝑁) ∈ ℝ) |
16 | 1 | nnrpd 12417 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
17 | 5 | nnzd 12074 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
18 | 16, 17 | rpexpcld 13604 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
19 | 15, 18 | ltsubrpd 12451 | . . 3 ⊢ (𝜑 → ((𝐶↑𝑁) − (𝐴↑𝑁)) < (𝐶↑𝑁)) |
20 | 12, 19 | eqbrtrd 5052 | . 2 ⊢ (𝜑 → (𝐵↑𝑁) < (𝐶↑𝑁)) |
21 | 8 | nnrpd 12417 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
22 | 13 | nnrpd 12417 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
23 | 21, 22, 5 | ltexp1d 39498 | . 2 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐵↑𝑁) < (𝐶↑𝑁))) |
24 | 20, 23 | mpbird 260 | 1 ⊢ (𝜑 → 𝐵 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 + caddc 10529 < clt 10664 − cmin 10859 ℕcn 11625 3c3 11681 ℤ≥cuz 12231 ↑cexp 13425 |
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-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 |
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-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 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-seq 13365 df-exp 13426 |
This theorem is referenced by: fltnltalem 39618 fltnlta 39619 |
Copyright terms: Public domain | W3C validator |