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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 7438 | . . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1)) | |
2 | oveq2 7439 | . . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1)) | |
3 | 1, 2 | eqeq12d 2751 | . 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1))) |
4 | oveq1 7438 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
5 | oveq2 7439 | . . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦)) | |
6 | 4, 5 | eqeq12d 2751 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦))) |
7 | oveq1 7438 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) | |
8 | oveq2 7439 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1))) | |
9 | 7, 8 | eqeq12d 2751 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
10 | oveq1 7438 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
11 | oveq2 7439 | . . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴)) | |
12 | 10, 11 | eqeq12d 2751 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴))) |
13 | eqid 2735 | . 2 ⊢ (1 + 1) = (1 + 1) | |
14 | oveq1 7438 | . . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1)) | |
15 | 1cnd 11254 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ) | |
16 | nncn 12272 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
17 | 15, 16, 15 | addassd 11281 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1))) |
18 | 14, 17 | sylan9eqr 2797 | . . 3 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))) |
19 | 18 | ex 412 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
20 | 3, 6, 9, 12, 13, 19 | nnind 12282 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 1c1 11154 + caddc 11156 ℕcn 12264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-addcl 11213 ax-addass 11218 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 |
This theorem is referenced by: nnaddcom 42282 renegmulnnass 42460 |
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