![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0addcom | Structured version Visualization version GIF version |
Description: Addition is commutative for nonnegative integers. Proven without ax-mulcom 11202. (Contributed by SN, 1-Feb-2025.) |
Ref | Expression |
---|---|
nn0addcom | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12504 | . 2 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) | |
2 | elnn0 12504 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
3 | nnaddcom 41908 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
4 | nnre 12249 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
5 | readdlid 42023 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (0 + 𝐵) = 𝐵) | |
6 | readdrid 42029 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵 + 0) = 𝐵) | |
7 | 5, 6 | eqtr4d 2768 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (0 + 𝐵) = (𝐵 + 0)) |
8 | 4, 7 | syl 17 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 + 𝐵) = (𝐵 + 0)) |
9 | oveq1 7423 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 𝐵) = (0 + 𝐵)) | |
10 | oveq2 7424 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐵 + 𝐴) = (𝐵 + 0)) | |
11 | 9, 10 | eqeq12d 2741 | . . . . . . 7 ⊢ (𝐴 = 0 → ((𝐴 + 𝐵) = (𝐵 + 𝐴) ↔ (0 + 𝐵) = (𝐵 + 0))) |
12 | 8, 11 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → (𝐴 = 0 → (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
13 | 12 | impcom 406 | . . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
14 | 3, 13 | jaoian 954 | . . . 4 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
15 | 2, 14 | sylanb 579 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
16 | nn0re 12511 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
17 | readdrid 42029 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) | |
18 | readdlid 42023 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) | |
19 | 17, 18 | eqtr4d 2768 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = (0 + 𝐴)) |
20 | 16, 19 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 0) = (0 + 𝐴)) |
21 | oveq2 7424 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐴 + 𝐵) = (𝐴 + 0)) | |
22 | oveq1 7423 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 + 𝐴) = (0 + 𝐴)) | |
23 | 21, 22 | eqeq12d 2741 | . . . . 5 ⊢ (𝐵 = 0 → ((𝐴 + 𝐵) = (𝐵 + 𝐴) ↔ (𝐴 + 0) = (0 + 𝐴))) |
24 | 20, 23 | syl5ibrcom 246 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → (𝐵 = 0 → (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
25 | 24 | imp 405 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
26 | 15, 25 | jaodan 955 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
27 | 1, 26 | sylan2b 592 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 (class class class)co 7416 ℝcr 11137 0cc0 11138 + caddc 11141 ℕcn 12242 ℕ0cn0 12502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-resub 41986 |
This theorem is referenced by: zaddcomlem 42071 zaddcom 42072 |
Copyright terms: Public domain | W3C validator |