decsucc - Metamath Proof Explorer

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Theorem decsucc 12719

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 12497	. . 3 ⊢ 9 ∈ ℕ₀
2		9p1e10 12680	. . . 4 ⊢ (9 + 1) = ;10
3	2	eqcomi 2735	. . 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 12681	. . . 4 ⊢ ;𝐴9 = ((;10 · 𝐴) + 9)
8	6, 7	eqtri 2754	. . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 9)
9	1, 3, 4, 5, 8	numsucc 12718	. 2 ⊢ (𝑁 + 1) = ((;10 · 𝐵) + 0)
10		dfdec10 12681	. 2 ⊢ ;𝐵0 = ((;10 · 𝐵) + 0)
11	9, 10	eqtr4i 2757	1 ⊢ (𝑁 + 1) = ;𝐵0

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7404 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 9c9 12275 ℕ₀cn0 12473 cdc 12678

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-dec 12679

This theorem is referenced by: sq10e99m1 14228 1259lem3 17073 1259lem4 17074 1259lem5 17075 2503lem2 17078 sqdeccom12 41740 fmtno5lem3 46776 tgoldbachlt 47037

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