declti - Metamath Proof Explorer

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Theorem declti 12721

Description: Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)

**Hypotheses**
Ref	Expression
declti.a	⊢ 𝐴 ∈ ℕ
declti.b	⊢ 𝐵 ∈ ℕ₀
declti.c	⊢ 𝐶 ∈ ℕ₀
declti.l	⊢ 𝐶 < ;10

**Assertion**
Ref	Expression
declti	⊢ 𝐶 < ;𝐴𝐵

**Proof of Theorem declti**
Step	Hyp	Ref	Expression
1		10nn 12699	. . 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 12720	. 2 ⊢ 𝐶 < ((;10 · 𝐴) + 𝐵)
7		dfdec10 12686	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
8	6, 7	breqtrri 5176	1 ⊢ 𝐶 < ;𝐴𝐵

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2104 class class class wbr 5149 (class class class)co 7413 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 < clt 11254 ℕcn 12218 ℕ₀cn0 12478 cdc 12683

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684

This theorem is referenced by: decltdi 12722 fsumcube 16010 5prm 17048 7prm 17050 11prm 17054 13prm 17055 17prm 17056 19prm 17057 23prm 17058 37prm 17060 43prm 17061 83prm 17062 139prm 17063 163prm 17064 317prm 17065 631prm 17066 1259lem5 17074 2503prm 17079 4001prm 17084 basendxnocndx 17334 basendxltdsndx 17339 dsndxnplusgndx 17341 dsndxnmulrndx 17342 slotsdnscsi 17343 dsndxntsetndx 17344 slotsdifdsndx 17345 basendxltunifndx 17349 unifndxntsetndx 17351 slotsdifunifndx 17352 slotsbhcdif 17366 slotsbhcdifOLD 17367 oppcbasOLD 17670 rescbasOLD 17783 rescabsOLD 17789 catstr 17915 mgpdsOLD 20044 sradsOLD 20954 thlbasOLD 21471 tuslemOLD 23994 setsmsdsOLD 24206 tmslemOLD 24213 tnglemOLD 24372 tngdsOLD 24387 log2le1 26689 bpos1 27020 bposlem9 27029 slotsinbpsd 27957 slotslnbpsd 27958 trkgstr 27960 ttgbasOLD 28396 ttgplusgOLD 28398 ttgvscaOLD 28401 eengstr 28503 basendxltedgfndx 28518 baseltedgfOLD 28519 zlmdsOLD 33239 hgt750lem 33959 257prm 46529 fmtno4prmfac193 46541 fmtno5nprm 46551 139prmALT 46564 127prm 46567 tgblthelfgott 46783

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