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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 11654 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | fltne.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
4 | eluzge3nn 12291 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11956 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | 2, 6 | expcld 13511 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
8 | fltne.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | 8 | nncnd 11654 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | 9, 6 | expcld 13511 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℂ) |
11 | fltne.1 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
12 | 7, 10, 11 | mvlladdd 11051 | . . 3 ⊢ (𝜑 → (𝐵↑𝑁) = ((𝐶↑𝑁) − (𝐴↑𝑁))) |
13 | fltne.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
14 | 13 | nnred 11653 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
15 | 14, 6 | reexpcld 13528 | . . . 4 ⊢ (𝜑 → (𝐶↑𝑁) ∈ ℝ) |
16 | 1 | nnrpd 12430 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
17 | 5 | nnzd 12087 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
18 | 16, 17 | rpexpcld 13609 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
19 | 15, 18 | ltsubrpd 12464 | . . 3 ⊢ (𝜑 → ((𝐶↑𝑁) − (𝐴↑𝑁)) < (𝐶↑𝑁)) |
20 | 12, 19 | eqbrtrd 5088 | . 2 ⊢ (𝜑 → (𝐵↑𝑁) < (𝐶↑𝑁)) |
21 | 8 | nnrpd 12430 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
22 | 13 | nnrpd 12430 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
23 | 21, 22, 5 | ltexp1d 39239 | . 2 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐵↑𝑁) < (𝐶↑𝑁))) |
24 | 20, 23 | mpbird 259 | 1 ⊢ (𝜑 → 𝐵 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 + caddc 10540 < clt 10675 − cmin 10870 ℕcn 11638 3c3 11694 ℤ≥cuz 12244 ↑cexp 13430 |
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-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 |
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-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 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-seq 13371 df-exp 13431 |
This theorem is referenced by: fltnltalem 39323 fltnlta 39324 |
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