![]() |
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 11176. (Contributed by SN, 1-Feb-2025.) |
Ref | Expression |
---|---|
nn0addcom | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12478 | . 2 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) | |
2 | elnn0 12478 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
3 | nnaddcom 41739 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
4 | nnre 12223 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
5 | readdlid 41854 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (0 + 𝐵) = 𝐵) | |
6 | readdrid 41860 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵 + 0) = 𝐵) | |
7 | 5, 6 | eqtr4d 2769 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (0 + 𝐵) = (𝐵 + 0)) |
8 | 4, 7 | syl 17 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 + 𝐵) = (𝐵 + 0)) |
9 | oveq1 7412 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 𝐵) = (0 + 𝐵)) | |
10 | oveq2 7413 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐵 + 𝐴) = (𝐵 + 0)) | |
11 | 9, 10 | eqeq12d 2742 | . . . . . . 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 407 | . . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
14 | 3, 13 | jaoian 953 | . . . 4 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
15 | 2, 14 | sylanb 580 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
16 | nn0re 12485 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
17 | readdrid 41860 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) | |
18 | readdlid 41854 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) | |
19 | 17, 18 | eqtr4d 2769 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = (0 + 𝐴)) |
20 | 16, 19 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 0) = (0 + 𝐴)) |
21 | oveq2 7413 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐴 + 𝐵) = (𝐴 + 0)) | |
22 | oveq1 7412 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 + 𝐴) = (0 + 𝐴)) | |
23 | 21, 22 | eqeq12d 2742 | . . . . 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 406 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
26 | 15, 25 | jaodan 954 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
27 | 1, 26 | sylan2b 593 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 (class class class)co 7405 ℝcr 11111 0cc0 11112 + caddc 11115 ℕcn 12216 ℕ0cn0 12476 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-resub 41817 |
This theorem is referenced by: zaddcomlem 41902 zaddcom 41903 |
Copyright terms: Public domain | W3C validator |