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Mirrors > Home > ILE Home > Th. List > decsuc | GIF version |
Description: The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
declt.a | ⊢ 𝐴 ∈ ℕ0 |
declt.b | ⊢ 𝐵 ∈ ℕ0 |
decsuc.c | ⊢ (𝐵 + 1) = 𝐶 |
decsuc.n | ⊢ 𝑁 = ;𝐴𝐵 |
Ref | Expression |
---|---|
decsuc | ⊢ (𝑁 + 1) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 9100 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | declt.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | declt.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | decsuc.c | . . 3 ⊢ (𝐵 + 1) = 𝐶 | |
5 | decsuc.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 9086 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2135 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | 1, 2, 3, 4, 7 | numsuc 9096 | . 2 ⊢ (𝑁 + 1) = ((;10 · 𝐴) + 𝐶) |
9 | dfdec10 9086 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
10 | 8, 9 | eqtr4i 2138 | 1 ⊢ (𝑁 + 1) = ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ∈ wcel 1463 (class class class)co 5728 0cc0 7544 1c1 7545 + caddc 7547 · cmul 7549 ℕ0cn0 8878 ;cdc 9083 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-setind 4412 ax-cnex 7633 ax-resscn 7634 ax-1cn 7635 ax-1re 7636 ax-icn 7637 ax-addcl 7638 ax-addrcl 7639 ax-mulcl 7640 ax-addcom 7642 ax-mulcom 7643 ax-addass 7644 ax-mulass 7645 ax-distr 7646 ax-i2m1 7647 ax-1rid 7649 ax-0id 7650 ax-rnegex 7651 ax-cnre 7653 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-sub 7855 df-inn 8628 df-2 8686 df-3 8687 df-4 8688 df-5 8689 df-6 8690 df-7 8691 df-8 8692 df-9 8693 df-n0 8879 df-dec 9084 |
This theorem is referenced by: 6p5lem 9152 ex-exp 12626 |
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