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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 11968 | . . 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 11977 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) < ((;10 · 𝐴) + 𝐶) |
7 | dfdec10 11955 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | dfdec10 11955 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
9 | 6, 7, 8 | 3brtr4i 4998 | 1 ⊢ ;𝐴𝐵 < ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2083 class class class wbr 4968 (class class class)co 7023 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 < clt 10528 ℕcn 11492 ℕ0cn0 11751 ;cdc 11952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-ltxr 10533 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-dec 11953 |
This theorem is referenced by: 23prm 16285 37prm 16287 43prm 16288 83prm 16289 163prm 16291 317prm 16292 1259prm 16302 2503lem3 16305 odrngstr 16512 slotsbhcdif 16526 imasvalstr 16558 prdsvalstr 16559 oppchomfval 16817 oppcbas 16821 rescco 16935 catstr 17060 ipostr 17596 cnfldstr 20233 cnfldfun 20243 thlle 20527 log2ub 25213 bpos1 25545 trkgstr 25916 ttgval 26348 ttglem 26349 ttgds 26354 eengstr 26453 hgt750lem 31535 257prm 43227 fmtno4nprmfac193 43240 31prm 43264 127prm 43267 evengpoap3 43468 nnsum4primesevenALTV 43470 tgblthelfgott 43484 |
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