![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > declti | Structured version Visualization version GIF version |
Description: Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
declti.a | ⊢ 𝐴 ∈ ℕ |
declti.b | ⊢ 𝐵 ∈ ℕ0 |
declti.c | ⊢ 𝐶 ∈ ℕ0 |
declti.l | ⊢ 𝐶 < ;10 |
Ref | Expression |
---|---|
declti | ⊢ 𝐶 < ;𝐴𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn 11925 | . . 3 ⊢ ;10 ∈ ℕ | |
2 | declti.a | . . 3 ⊢ 𝐴 ∈ ℕ | |
3 | declti.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | declti.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | declti.l | . . 3 ⊢ 𝐶 < ;10 | |
6 | 1, 2, 3, 4, 5 | numlti 11947 | . 2 ⊢ 𝐶 < ((;10 · 𝐴) + 𝐵) |
7 | dfdec10 11912 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | 6, 7 | breqtrri 4952 | 1 ⊢ 𝐶 < ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2051 class class class wbr 4925 (class class class)co 6974 0cc0 10333 1c1 10334 + caddc 10336 · cmul 10338 < clt 10472 ℕcn 11437 ℕ0cn0 11705 ;cdc 11909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 |
This theorem is referenced by: decltdi 11949 fsumcube 15272 5prm 16296 7prm 16298 11prm 16302 13prm 16303 17prm 16304 19prm 16305 23prm 16306 37prm 16308 43prm 16309 83prm 16310 139prm 16311 163prm 16312 317prm 16313 631prm 16314 1259lem5 16322 2503prm 16327 4001prm 16332 ressds 16540 resshom 16545 ressco 16546 slotsbhcdif 16547 oppcbas 16858 rescbas 16969 rescabs 16973 catstr 17097 mgpds 18984 srads 19692 thlbas 20557 ressunif 22589 tuslem 22594 setsmsds 22804 tmslem 22810 tnglem 22967 tngds 22975 log2le1 25245 bpos1 25576 bposlem9 25585 trkgstr 25947 ttgbas 26381 ttgplusg 26382 ttgvsca 26384 eengstr 26484 baseltedgf 26497 zlmds 30881 hgt750lem 31602 257prm 43125 fmtno4prmfac193 43137 fmtno5nprm 43147 139prmALT 43161 127prm 43165 tgblthelfgott 43382 |
Copyright terms: Public domain | W3C validator |