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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 12749 | . . 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 12770 | . 2 ⊢ 𝐶 < ((;10 · 𝐴) + 𝐵) |
| 7 | dfdec10 12736 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
| 8 | 6, 7 | breqtrri 5170 | 1 ⊢ 𝐶 < ;𝐴𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ℕcn 12266 ℕ0cn0 12526 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 |
| This theorem is referenced by: decltdi 12772 fsumcube 16096 5prm 17146 7prm 17148 11prm 17152 13prm 17153 17prm 17154 19prm 17155 23prm 17156 37prm 17158 43prm 17159 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem5 17172 2503prm 17177 4001prm 17182 basendxnocndx 17427 basendxltdsndx 17432 dsndxnplusgndx 17434 dsndxnmulrndx 17435 slotsdnscsi 17436 dsndxntsetndx 17437 slotsdifdsndx 17438 basendxltunifndx 17442 unifndxntsetndx 17444 slotsdifunifndx 17445 slotsbhcdif 17459 slotsbhcdifOLD 17460 rescabsOLD 17878 catstr 18005 sradsOLD 21192 thlbasOLD 21715 tuslemOLD 24276 setsmsdsOLD 24488 tmslemOLD 24495 tnglemOLD 24654 tngdsOLD 24669 log2le1 26993 bpos1 27327 bposlem9 27336 slotsinbpsd 28449 slotslnbpsd 28450 trkgstr 28452 ttgbasOLD 28888 ttgplusgOLD 28890 ttgvscaOLD 28893 eengstr 28995 basendxltedgfndx 29010 baseltedgfOLD 29011 zlmdsOLD 33962 hgt750lem 34666 257prm 47548 fmtno4prmfac193 47560 fmtno5nprm 47570 139prmALT 47583 127prm 47586 tgblthelfgott 47802 |
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