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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 7419 | . . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1)) | |
2 | oveq2 7420 | . . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1)) | |
3 | 1, 2 | eqeq12d 2747 | . 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1))) |
4 | oveq1 7419 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
5 | oveq2 7420 | . . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦)) | |
6 | 4, 5 | eqeq12d 2747 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦))) |
7 | oveq1 7419 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) | |
8 | oveq2 7420 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1))) | |
9 | 7, 8 | eqeq12d 2747 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))) |
10 | oveq1 7419 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
11 | oveq2 7420 | . . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴)) | |
12 | 10, 11 | eqeq12d 2747 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴))) |
13 | eqid 2731 | . 2 ⊢ (1 + 1) = (1 + 1) | |
14 | oveq1 7419 | . . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1)) | |
15 | 1cnd 11214 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ) | |
16 | nncn 12225 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
17 | 15, 16, 15 | addassd 11241 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1))) |
18 | 14, 17 | sylan9eqr 2793 | . . 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 12235 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 1c1 11115 + caddc 11117 ℕcn 12217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-1cn 11172 ax-addcl 11174 ax-addass 11179 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12218 |
This theorem is referenced by: nnaddcom 41485 renegmulnnass 41629 |
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