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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 7220 | . . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1)) | |
2 | oveq2 7221 | . . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1)) | |
3 | 1, 2 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1))) |
4 | oveq1 7220 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
5 | oveq2 7221 | . . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦)) | |
6 | 4, 5 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦))) |
7 | oveq1 7220 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) | |
8 | oveq2 7221 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1))) | |
9 | 7, 8 | eqeq12d 2753 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
10 | oveq1 7220 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
11 | oveq2 7221 | . . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴)) | |
12 | 10, 11 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴))) |
13 | eqid 2737 | . 2 ⊢ (1 + 1) = (1 + 1) | |
14 | oveq1 7220 | . . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1)) | |
15 | 1cnd 10828 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ) | |
16 | nncn 11838 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
17 | 15, 16, 15 | addassd 10855 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1))) |
18 | 14, 17 | sylan9eqr 2800 | . . 3 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))) |
19 | 18 | ex 416 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
20 | 3, 6, 9, 12, 13, 19 | nnind 11848 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 (class class class)co 7213 1c1 10730 + caddc 10732 ℕcn 11830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 ax-1cn 10787 ax-addcl 10789 ax-addass 10794 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-nn 11831 |
This theorem is referenced by: nnaddcom 40005 |
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