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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 11216. Since (𝐴 · 1) is 𝐴 by ax-1rid 11222, this is equivalent to remullid 42439 for natural numbers, but using fewer axioms (avoiding ax-resscn 11209, ax-addass 11217, ax-mulass 11218, ax-rnegex 11223, ax-pre-lttri 11226, ax-pre-lttrn 11227, ax-pre-ltadd 11228). (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
nnmul1com | ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7438 | . . . 4 ⊢ (𝑥 = 1 → (1 · 𝑥) = (1 · 1)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
3 | 1, 2 | eqeq12d 2750 | . . 3 ⊢ (𝑥 = 1 → ((1 · 𝑥) = 𝑥 ↔ (1 · 1) = 1)) |
4 | oveq2 7438 | . . . 4 ⊢ (𝑥 = 𝑦 → (1 · 𝑥) = (1 · 𝑦)) | |
5 | id 22 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2750 | . . 3 ⊢ (𝑥 = 𝑦 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝑦) = 𝑦)) |
7 | oveq2 7438 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1 · 𝑥) = (1 · (𝑦 + 1))) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → 𝑥 = (𝑦 + 1)) | |
9 | 7, 8 | eqeq12d 2750 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((1 · 𝑥) = 𝑥 ↔ (1 · (𝑦 + 1)) = (𝑦 + 1))) |
10 | oveq2 7438 | . . . 4 ⊢ (𝑥 = 𝐴 → (1 · 𝑥) = (1 · 𝐴)) | |
11 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2750 | . . 3 ⊢ (𝑥 = 𝐴 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝐴) = 𝐴)) |
13 | 1t1e1ALT 42274 | . . 3 ⊢ (1 · 1) = 1 | |
14 | 1cnd 11253 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 1 ∈ ℂ) | |
15 | simpl 482 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℕ) | |
16 | 15 | nncnd 12279 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℂ) |
17 | 14, 16, 14 | adddid 11282 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = ((1 · 𝑦) + (1 · 1))) |
18 | simpr 484 | . . . . . 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 7448 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → ((1 · 𝑦) + (1 · 1)) = (𝑦 + 1)) |
21 | 17, 20 | eqtrd 2774 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = (𝑦 + 1)) |
22 | 21 | ex 412 | . . 3 ⊢ (𝑦 ∈ ℕ → ((1 · 𝑦) = 𝑦 → (1 · (𝑦 + 1)) = (𝑦 + 1))) |
23 | 3, 6, 9, 12, 13, 22 | nnind 12281 | . 2 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = 𝐴) |
24 | nnre 12270 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
25 | ax-1rid 11222 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 23, 26 | eqtr4d 2777 | 1 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℝcr 11151 1c1 11153 + caddc 11155 · cmul 11157 ℕcn 12263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rrecex 11224 ax-cnre 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 |
This theorem is referenced by: nnmulcom 42285 |
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