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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnadd1com | Structured version Visualization version GIF version |
Description: Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.) |
Ref | Expression |
---|---|
nnadd1com | ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7411 | . . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1)) | |
2 | oveq2 7412 | . . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1)) | |
3 | 1, 2 | eqeq12d 2749 | . 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1))) |
4 | oveq1 7411 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
5 | oveq2 7412 | . . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦)) | |
6 | 4, 5 | eqeq12d 2749 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦))) |
7 | oveq1 7411 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) | |
8 | oveq2 7412 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1))) | |
9 | 7, 8 | eqeq12d 2749 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
10 | oveq1 7411 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
11 | oveq2 7412 | . . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴)) | |
12 | 10, 11 | eqeq12d 2749 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴))) |
13 | eqid 2733 | . 2 ⊢ (1 + 1) = (1 + 1) | |
14 | oveq1 7411 | . . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1)) | |
15 | 1cnd 11205 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ) | |
16 | nncn 12216 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
17 | 15, 16, 15 | addassd 11232 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1))) |
18 | 14, 17 | sylan9eqr 2795 | . . 3 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))) |
19 | 18 | ex 414 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
20 | 3, 6, 9, 12, 13, 19 | nnind 12226 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7404 1c1 11107 + caddc 11109 ℕcn 12208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 ax-1cn 11164 ax-addcl 11166 ax-addass 11171 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-nn 12209 |
This theorem is referenced by: nnaddcom 41132 renegmulnnass 41270 |
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