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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xralrple4 | Structured version Visualization version GIF version |
Description: Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
xralrple4.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xralrple4.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
xralrple4.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
xralrple4 | ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xralrple4.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | 1 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐴 ∈ ℝ*) |
3 | xralrple4.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 11210 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | 4 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℝ*) |
6 | 3 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℝ) |
7 | rpre 12928 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
8 | 7 | adantl 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
9 | xralrple4.n | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
10 | 9 | nnnn0d 12478 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
11 | 10 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑁 ∈ ℕ0) |
12 | 8, 11 | reexpcld 14074 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑁) ∈ ℝ) |
13 | 12 | adantlr 714 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑁) ∈ ℝ) |
14 | 6, 13 | readdcld 11189 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝐵 + (𝑥↑𝑁)) ∈ ℝ) |
15 | 14 | rexrd 11210 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝐵 + (𝑥↑𝑁)) ∈ ℝ*) |
16 | simplr 768 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐴 ≤ 𝐵) | |
17 | rpge0 12933 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 ≤ 𝑥) | |
18 | 17 | adantl 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 ≤ 𝑥) |
19 | 8, 11, 18 | expge0d 14075 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 ≤ (𝑥↑𝑁)) |
20 | 3 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℝ) |
21 | 20, 12 | addge01d 11748 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (0 ≤ (𝑥↑𝑁) ↔ 𝐵 ≤ (𝐵 + (𝑥↑𝑁)))) |
22 | 19, 21 | mpbid 231 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐵 ≤ (𝐵 + (𝑥↑𝑁))) |
23 | 22 | adantlr 714 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐵 ≤ (𝐵 + (𝑥↑𝑁))) |
24 | 2, 5, 15, 16, 23 | xrletrd 13087 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) |
25 | 24 | ralrimiva 3140 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) |
26 | 25 | ex 414 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁)))) |
27 | simpr 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+) | |
28 | 9 | nnrpd 12960 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ+) |
29 | 28 | rpreccld 12972 | . . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ+) |
30 | 29 | rpred 12962 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
31 | 30 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑁) ∈ ℝ) |
32 | 27, 31 | rpcxpcld 26103 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦↑𝑐(1 / 𝑁)) ∈ ℝ+) |
33 | 32 | adantlr 714 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) ∧ 𝑦 ∈ ℝ+) → (𝑦↑𝑐(1 / 𝑁)) ∈ ℝ+) |
34 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) ∧ 𝑦 ∈ ℝ+) → ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) | |
35 | oveq1 7365 | . . . . . . . . . 10 ⊢ (𝑥 = (𝑦↑𝑐(1 / 𝑁)) → (𝑥↑𝑁) = ((𝑦↑𝑐(1 / 𝑁))↑𝑁)) | |
36 | 35 | oveq2d 7374 | . . . . . . . . 9 ⊢ (𝑥 = (𝑦↑𝑐(1 / 𝑁)) → (𝐵 + (𝑥↑𝑁)) = (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁))) |
37 | 36 | breq2d 5118 | . . . . . . . 8 ⊢ (𝑥 = (𝑦↑𝑐(1 / 𝑁)) → (𝐴 ≤ (𝐵 + (𝑥↑𝑁)) ↔ 𝐴 ≤ (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁)))) |
38 | 37 | rspcva 3578 | . . . . . . 7 ⊢ (((𝑦↑𝑐(1 / 𝑁)) ∈ ℝ+ ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) → 𝐴 ≤ (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁))) |
39 | 33, 34, 38 | syl2anc 585 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) ∧ 𝑦 ∈ ℝ+) → 𝐴 ≤ (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁))) |
40 | 27 | rpcnd 12964 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℂ) |
41 | 9 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑁 ∈ ℕ) |
42 | cxproot 26061 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑦↑𝑐(1 / 𝑁))↑𝑁) = 𝑦) | |
43 | 40, 41, 42 | syl2anc 585 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝑦↑𝑐(1 / 𝑁))↑𝑁) = 𝑦) |
44 | 43 | oveq2d 7374 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁)) = (𝐵 + 𝑦)) |
45 | 44 | adantlr 714 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) ∧ 𝑦 ∈ ℝ+) → (𝐵 + ((𝑦↑𝑐(1 / 𝑁))↑𝑁)) = (𝐵 + 𝑦)) |
46 | 39, 45 | breqtrd 5132 | . . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) ∧ 𝑦 ∈ ℝ+) → 𝐴 ≤ (𝐵 + 𝑦)) |
47 | 46 | ralrimiva 3140 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) → ∀𝑦 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑦)) |
48 | xralrple 13130 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑦))) | |
49 | 1, 3, 48 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑦))) |
50 | 49 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑦))) |
51 | 47, 50 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁))) → 𝐴 ≤ 𝐵) |
52 | 51 | ex 414 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁)) → 𝐴 ≤ 𝐵)) |
53 | 26, 52 | impbid 211 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 class class class wbr 5106 (class class class)co 7358 ℂcc 11054 ℝcr 11055 0cc0 11056 1c1 11057 + caddc 11059 ℝ*cxr 11193 ≤ cle 11195 / cdiv 11817 ℕcn 12158 ℕ0cn0 12418 ℝ+crp 12920 ↑cexp 13973 ↑𝑐ccxp 25927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-shft 14958 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-ef 15955 df-sin 15957 df-cos 15958 df-pi 15960 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-cxp 25929 |
This theorem is referenced by: (None) |
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