![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nnmul1com | Structured version Visualization version GIF version |
Description: Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11170. Since (𝐴 · 1) is 𝐴 by ax-1rid 11176, this is equivalent to remullid 41302 for natural numbers, but using fewer axioms (avoiding ax-resscn 11163, ax-addass 11171, ax-mulass 11172, ax-rnegex 11177, ax-pre-lttri 11180, ax-pre-lttrn 11181, ax-pre-ltadd 11182). (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
nnmul1com | ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7413 | . . . 4 ⊢ (𝑥 = 1 → (1 · 𝑥) = (1 · 1)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
3 | 1, 2 | eqeq12d 2748 | . . 3 ⊢ (𝑥 = 1 → ((1 · 𝑥) = 𝑥 ↔ (1 · 1) = 1)) |
4 | oveq2 7413 | . . . 4 ⊢ (𝑥 = 𝑦 → (1 · 𝑥) = (1 · 𝑦)) | |
5 | id 22 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2748 | . . 3 ⊢ (𝑥 = 𝑦 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝑦) = 𝑦)) |
7 | oveq2 7413 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1 · 𝑥) = (1 · (𝑦 + 1))) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → 𝑥 = (𝑦 + 1)) | |
9 | 7, 8 | eqeq12d 2748 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((1 · 𝑥) = 𝑥 ↔ (1 · (𝑦 + 1)) = (𝑦 + 1))) |
10 | oveq2 7413 | . . . 4 ⊢ (𝑥 = 𝐴 → (1 · 𝑥) = (1 · 𝐴)) | |
11 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2748 | . . 3 ⊢ (𝑥 = 𝐴 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝐴) = 𝐴)) |
13 | 1t1e1ALT 41173 | . . 3 ⊢ (1 · 1) = 1 | |
14 | 1cnd 11205 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 1 ∈ ℂ) | |
15 | simpl 483 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℕ) | |
16 | 15 | nncnd 12224 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℂ) |
17 | 14, 16, 14 | adddid 11234 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = ((1 · 𝑦) + (1 · 1))) |
18 | simpr 485 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · 𝑦) = 𝑦) | |
19 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · 1) = 1) |
20 | 18, 19 | oveq12d 7423 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → ((1 · 𝑦) + (1 · 1)) = (𝑦 + 1)) |
21 | 17, 20 | eqtrd 2772 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = (𝑦 + 1)) |
22 | 21 | ex 413 | . . 3 ⊢ (𝑦 ∈ ℕ → ((1 · 𝑦) = 𝑦 → (1 · (𝑦 + 1)) = (𝑦 + 1))) |
23 | 3, 6, 9, 12, 13, 22 | nnind 12226 | . 2 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = 𝐴) |
24 | nnre 12215 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
25 | ax-1rid 11176 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 23, 26 | eqtr4d 2775 | 1 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 (class class class)co 7405 ℝcr 11105 1c1 11107 + caddc 11109 · cmul 11111 ℕcn 12208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rrecex 11178 ax-cnre 11179 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12209 |
This theorem is referenced by: nnmulcom 41183 |
Copyright terms: Public domain | W3C validator |