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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 10681. Since (𝐴 · 1) is 𝐴 by ax-1rid 10687, this is equivalent to remulid2 40014 for natural numbers, but using fewer axioms (avoiding ax-resscn 10674, ax-addass 10682, ax-mulass 10683, ax-rnegex 10688, ax-pre-lttri 10691, ax-pre-lttrn 10692, ax-pre-ltadd 10693). (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
nnmul1com | ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7180 | . . . 4 ⊢ (𝑥 = 1 → (1 · 𝑥) = (1 · 1)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
3 | 1, 2 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = 1 → ((1 · 𝑥) = 𝑥 ↔ (1 · 1) = 1)) |
4 | oveq2 7180 | . . . 4 ⊢ (𝑥 = 𝑦 → (1 · 𝑥) = (1 · 𝑦)) | |
5 | id 22 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = 𝑦 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝑦) = 𝑦)) |
7 | oveq2 7180 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1 · 𝑥) = (1 · (𝑦 + 1))) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → 𝑥 = (𝑦 + 1)) | |
9 | 7, 8 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((1 · 𝑥) = 𝑥 ↔ (1 · (𝑦 + 1)) = (𝑦 + 1))) |
10 | oveq2 7180 | . . . 4 ⊢ (𝑥 = 𝐴 → (1 · 𝑥) = (1 · 𝐴)) | |
11 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = 𝐴 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝐴) = 𝐴)) |
13 | 1t1e1ALT 39890 | . . 3 ⊢ (1 · 1) = 1 | |
14 | 1cnd 10716 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 1 ∈ ℂ) | |
15 | simpl 486 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℕ) | |
16 | 15 | nncnd 11734 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℂ) |
17 | 14, 16, 14 | adddid 10745 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = ((1 · 𝑦) + (1 · 1))) |
18 | simpr 488 | . . . . . 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 7190 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → ((1 · 𝑦) + (1 · 1)) = (𝑦 + 1)) |
21 | 17, 20 | eqtrd 2773 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = (𝑦 + 1)) |
22 | 21 | ex 416 | . . 3 ⊢ (𝑦 ∈ ℕ → ((1 · 𝑦) = 𝑦 → (1 · (𝑦 + 1)) = (𝑦 + 1))) |
23 | 3, 6, 9, 12, 13, 22 | nnind 11736 | . 2 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = 𝐴) |
24 | nnre 11725 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
25 | ax-1rid 10687 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 23, 26 | eqtr4d 2776 | 1 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 (class class class)co 7172 ℝcr 10616 1c1 10618 + caddc 10620 · cmul 10622 ℕcn 11718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rrecex 10689 ax-cnre 10690 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7175 df-om 7602 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-nn 11719 |
This theorem is referenced by: nnmulcom 39900 |
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