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Mirrors > Home > MPE Home > Th. List > declt | Structured version Visualization version GIF version |
Description: Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
declt.a | ⊢ 𝐴 ∈ ℕ0 |
declt.b | ⊢ 𝐵 ∈ ℕ0 |
declt.c | ⊢ 𝐶 ∈ ℕ |
declt.l | ⊢ 𝐵 < 𝐶 |
Ref | Expression |
---|---|
declt | ⊢ ;𝐴𝐵 < ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn 11706 | . . 3 ⊢ ;10 ∈ ℕ | |
2 | declt.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | declt.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | declt.c | . . 3 ⊢ 𝐶 ∈ ℕ | |
5 | declt.l | . . 3 ⊢ 𝐵 < 𝐶 | |
6 | 1, 2, 3, 4, 5 | numlt 11719 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) < ((;10 · 𝐴) + 𝐶) |
7 | dfdec10 11689 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | dfdec10 11689 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
9 | 6, 7, 8 | 3brtr4i 4834 | 1 ⊢ ;𝐴𝐵 < ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 class class class wbr 4804 (class class class)co 6813 0cc0 10128 1c1 10129 + caddc 10131 · cmul 10133 < clt 10266 ℕcn 11212 ℕ0cn0 11484 ;cdc 11685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-ltxr 10271 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-dec 11686 |
This theorem is referenced by: 23prm 16028 37prm 16030 43prm 16031 83prm 16032 163prm 16034 317prm 16035 1259prm 16045 2503lem3 16048 odrngstr 16268 slotsbhcdif 16282 imasvalstr 16314 prdsvalstr 16315 oppchomfval 16575 oppcbas 16579 rescco 16693 catstr 16818 ipostr 17354 cnfldstr 19950 cnfldfun 19960 thlle 20243 log2ub 24875 bpos1 25207 trkgstr 25542 ttgval 25954 ttglem 25955 ttgds 25960 eengstr 26059 hgt750lem 31038 257prm 41983 fmtno4nprmfac193 41996 31prm 42022 127prm 42025 evengpoap3 42197 nnsum4primesevenALTV 42199 tgblthelfgott 42213 tgblthelfgottOLD 42219 |
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