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Mirrors > Home > MPE Home > Th. List > numsucc | Structured version Visualization version GIF version |
Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numsucc.1 | โข ๐ โ โ0 |
numsucc.2 | โข ๐ = (๐ + 1) |
numsucc.3 | โข ๐ด โ โ0 |
numsucc.4 | โข (๐ด + 1) = ๐ต |
numsucc.5 | โข ๐ = ((๐ ยท ๐ด) + ๐) |
Ref | Expression |
---|---|
numsucc | โข (๐ + 1) = ((๐ ยท ๐ต) + 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsucc.2 | . . . . . . 7 โข ๐ = (๐ + 1) | |
2 | numsucc.1 | . . . . . . . 8 โข ๐ โ โ0 | |
3 | 1nn0 12430 | . . . . . . . 8 โข 1 โ โ0 | |
4 | 2, 3 | nn0addcli 12451 | . . . . . . 7 โข (๐ + 1) โ โ0 |
5 | 1, 4 | eqeltri 2834 | . . . . . 6 โข ๐ โ โ0 |
6 | 5 | nn0cni 12426 | . . . . 5 โข ๐ โ โ |
7 | 6 | mulid1i 11160 | . . . 4 โข (๐ ยท 1) = ๐ |
8 | 7 | oveq2i 7369 | . . 3 โข ((๐ ยท ๐ด) + (๐ ยท 1)) = ((๐ ยท ๐ด) + ๐) |
9 | numsucc.3 | . . . . 5 โข ๐ด โ โ0 | |
10 | 9 | nn0cni 12426 | . . . 4 โข ๐ด โ โ |
11 | ax-1cn 11110 | . . . 4 โข 1 โ โ | |
12 | 6, 10, 11 | adddii 11168 | . . 3 โข (๐ ยท (๐ด + 1)) = ((๐ ยท ๐ด) + (๐ ยท 1)) |
13 | 1 | eqcomi 2746 | . . . 4 โข (๐ + 1) = ๐ |
14 | numsucc.5 | . . . 4 โข ๐ = ((๐ ยท ๐ด) + ๐) | |
15 | 5, 9, 2, 13, 14 | numsuc 12633 | . . 3 โข (๐ + 1) = ((๐ ยท ๐ด) + ๐) |
16 | 8, 12, 15 | 3eqtr4ri 2776 | . 2 โข (๐ + 1) = (๐ ยท (๐ด + 1)) |
17 | numsucc.4 | . . 3 โข (๐ด + 1) = ๐ต | |
18 | 17 | oveq2i 7369 | . 2 โข (๐ ยท (๐ด + 1)) = (๐ ยท ๐ต) |
19 | 9, 3 | nn0addcli 12451 | . . . 4 โข (๐ด + 1) โ โ0 |
20 | 17, 19 | eqeltrri 2835 | . . 3 โข ๐ต โ โ0 |
21 | 5, 20 | num0u 12630 | . 2 โข (๐ ยท ๐ต) = ((๐ ยท ๐ต) + 0) |
22 | 16, 18, 21 | 3eqtri 2769 | 1 โข (๐ + 1) = ((๐ ยท ๐ต) + 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7358 0cc0 11052 1c1 11053 + caddc 11055 ยท cmul 11057 โ0cn0 12414 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-nn 12155 df-n0 12415 |
This theorem is referenced by: decsucc 12660 |
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