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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 9418 | . . 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 9404 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2209 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | 1, 2, 3, 4, 7 | numsuc 9414 | . 2 ⊢ (𝑁 + 1) = ((;10 · 𝐴) + 𝐶) |
9 | dfdec10 9404 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
10 | 8, 9 | eqtr4i 2212 | 1 ⊢ (𝑁 + 1) = ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2159 (class class class)co 5890 0cc0 7828 1c1 7829 + caddc 7831 · cmul 7833 ℕ0cn0 9193 ;cdc 9401 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-sub 8147 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-7 9000 df-8 9001 df-9 9002 df-n0 9194 df-dec 9402 |
This theorem is referenced by: 6p5lem 9470 ex-exp 14862 |
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