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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 12726 | . . 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 12747 | . 2 ⊢ 𝐶 < ((;10 · 𝐴) + 𝐵) |
7 | dfdec10 12713 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | 6, 7 | breqtrri 5176 | 1 ⊢ 𝐶 < ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 class class class wbr 5149 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 < clt 11280 ℕcn 12245 ℕ0cn0 12505 ;cdc 12710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 |
This theorem is referenced by: decltdi 12749 fsumcube 16040 5prm 17081 7prm 17083 11prm 17087 13prm 17088 17prm 17089 19prm 17090 23prm 17091 37prm 17093 43prm 17094 83prm 17095 139prm 17096 163prm 17097 317prm 17098 631prm 17099 1259lem5 17107 2503prm 17112 4001prm 17117 basendxnocndx 17367 basendxltdsndx 17372 dsndxnplusgndx 17374 dsndxnmulrndx 17375 slotsdnscsi 17376 dsndxntsetndx 17377 slotsdifdsndx 17378 basendxltunifndx 17382 unifndxntsetndx 17384 slotsdifunifndx 17385 slotsbhcdif 17399 slotsbhcdifOLD 17400 oppcbasOLD 17703 rescbasOLD 17816 rescabsOLD 17822 catstr 17951 mgpdsOLD 20100 sradsOLD 21090 thlbasOLD 21646 tuslemOLD 24216 setsmsdsOLD 24428 tmslemOLD 24435 tnglemOLD 24594 tngdsOLD 24609 log2le1 26927 bpos1 27261 bposlem9 27270 slotsinbpsd 28317 slotslnbpsd 28318 trkgstr 28320 ttgbasOLD 28756 ttgplusgOLD 28758 ttgvscaOLD 28761 eengstr 28863 basendxltedgfndx 28878 baseltedgfOLD 28879 zlmdsOLD 33695 hgt750lem 34414 257prm 47038 fmtno4prmfac193 47050 fmtno5nprm 47060 139prmALT 47073 127prm 47076 tgblthelfgott 47292 |
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