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| Mirrors > Home > ILE Home > Th. List > numsucc | Unicode version | ||
| Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| Ref | Expression |
|---|---|
| numsucc.1 |
|
| numsucc.2 |
|
| numsucc.3 |
|
| numsucc.4 |
|
| numsucc.5 |
|
| Ref | Expression |
|---|---|
| numsucc |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | numsucc.2 |
. . . . . . 7
| |
| 2 | numsucc.1 |
. . . . . . . 8
| |
| 3 | 1nn0 9311 |
. . . . . . . 8
| |
| 4 | 2, 3 | nn0addcli 9332 |
. . . . . . 7
|
| 5 | 1, 4 | eqeltri 2278 |
. . . . . 6
|
| 6 | 5 | nn0cni 9307 |
. . . . 5
|
| 7 | 6 | mulridi 8074 |
. . . 4
|
| 8 | 7 | oveq2i 5955 |
. . 3
|
| 9 | numsucc.3 |
. . . . 5
| |
| 10 | 9 | nn0cni 9307 |
. . . 4
|
| 11 | ax-1cn 8018 |
. . . 4
| |
| 12 | 6, 10, 11 | adddii 8082 |
. . 3
|
| 13 | 1 | eqcomi 2209 |
. . . 4
|
| 14 | numsucc.5 |
. . . 4
| |
| 15 | 5, 9, 2, 13, 14 | numsuc 9517 |
. . 3
|
| 16 | 8, 12, 15 | 3eqtr4ri 2237 |
. 2
|
| 17 | numsucc.4 |
. . 3
| |
| 18 | 17 | oveq2i 5955 |
. 2
|
| 19 | 9, 3 | nn0addcli 9332 |
. . . 4
|
| 20 | 17, 19 | eqeltrri 2279 |
. . 3
|
| 21 | 5, 20 | num0u 9514 |
. 2
|
| 22 | 16, 18, 21 | 3eqtri 2230 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-sub 8245 df-inn 9037 df-n0 9296 |
| This theorem is referenced by: decsucc 9544 |
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