decsucc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > decsucc		Structured version Visualization version GIF version

Theorem decsucc 12718

Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)

**Hypotheses**
Ref	Expression
decsucc.1	⊢ 𝐴 ∈ ℕ₀
decsucc.2	⊢ (𝐴 + 1) = 𝐵
decsucc.3	⊢ 𝑁 = ;𝐴9

**Assertion**
Ref	Expression
decsucc	⊢ (𝑁 + 1) = ;𝐵0

**Proof of Theorem decsucc**
Step	Hyp	Ref	Expression
1		9nn0 12496	. . 3 ⊢ 9 ∈ ℕ₀
2		9p1e10 12679	. . . 4 ⊢ (9 + 1) = ;10
3	2	eqcomi 2742	. . 3 ⊢ ;10 = (9 + 1)
4		decsucc.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
5		decsucc.2	. . 3 ⊢ (𝐴 + 1) = 𝐵
6		decsucc.3	. . . 4 ⊢ 𝑁 = ;𝐴9
7		dfdec10 12680	. . . 4 ⊢ ;𝐴9 = ((;10 · 𝐴) + 9)
8	6, 7	eqtri 2761	. . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 9)
9	1, 3, 4, 5, 8	numsucc 12717	. 2 ⊢ (𝑁 + 1) = ((;10 · 𝐵) + 0)
10		dfdec10 12680	. 2 ⊢ ;𝐵0 = ((;10 · 𝐵) + 0)
11	9, 10	eqtr4i 2764	1 ⊢ (𝑁 + 1) = ;𝐵0

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 9c9 12274 ℕ₀cn0 12472 cdc 12677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-dec 12678

This theorem is referenced by: sq10e99m1 14225 1259lem3 17066 1259lem4 17067 1259lem5 17068 2503lem2 17071 sqdeccom12 41201 fmtno5lem3 46223 tgoldbachlt 46484

W3C validator