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Mirrors > Home > ILE Home > Th. List > decaddi | GIF version |
Description: Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
decaddi.1 | ⊢ 𝐴 ∈ ℕ0 |
decaddi.2 | ⊢ 𝐵 ∈ ℕ0 |
decaddi.3 | ⊢ 𝑁 ∈ ℕ0 |
decaddi.4 | ⊢ 𝑀 = ;𝐴𝐵 |
decaddi.5 | ⊢ (𝐵 + 𝑁) = 𝐶 |
Ref | Expression |
---|---|
decaddi | ⊢ (𝑀 + 𝑁) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decaddi.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decaddi.2 | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 8406 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 8615 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 8403 | . . 3 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addid1i 7353 | . 2 ⊢ (𝐴 + 0) = 𝐴 |
9 | decaddi.5 | . 2 ⊢ (𝐵 + 𝑁) = 𝐶 | |
10 | 1, 2, 3, 4, 5, 6, 8, 9 | decadd 8647 | 1 ⊢ (𝑀 + 𝑁) = ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∈ wcel 1434 (class class class)co 5564 0cc0 7079 + caddc 7082 ℕ0cn0 8391 ;cdc 8594 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-setind 4309 ax-cnex 7165 ax-resscn 7166 ax-1cn 7167 ax-1re 7168 ax-icn 7169 ax-addcl 7170 ax-addrcl 7171 ax-mulcl 7172 ax-addcom 7174 ax-mulcom 7175 ax-addass 7176 ax-mulass 7177 ax-distr 7178 ax-i2m1 7179 ax-1rid 7181 ax-0id 7182 ax-rnegex 7183 ax-cnre 7185 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-br 3807 df-opab 3861 df-id 4077 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-iota 4918 df-fun 4955 df-fv 4961 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-sub 7384 df-inn 8143 df-2 8201 df-3 8202 df-4 8203 df-5 8204 df-6 8205 df-7 8206 df-8 8207 df-9 8208 df-n0 8392 df-dec 8595 |
This theorem is referenced by: 4t4e16 8692 6t3e18 8698 7t4e28 8704 7t7e49 8707 ex-fac 10809 |
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