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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 10601. Since (𝐴 · 1) is 𝐴 by ax-1rid 10607, this is equivalent to remulid2 39298 for natural numbers, but using fewer axioms (avoiding ax-resscn 10594, ax-addass 10602, ax-mulass 10603, ax-rnegex 10608, ax-pre-lttri 10611, ax-pre-lttrn 10612, ax-pre-ltadd 10613). (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
nnmul1com | ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7164 | . . . 4 ⊢ (𝑥 = 1 → (1 · 𝑥) = (1 · 1)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
3 | 1, 2 | eqeq12d 2837 | . . 3 ⊢ (𝑥 = 1 → ((1 · 𝑥) = 𝑥 ↔ (1 · 1) = 1)) |
4 | oveq2 7164 | . . . 4 ⊢ (𝑥 = 𝑦 → (1 · 𝑥) = (1 · 𝑦)) | |
5 | id 22 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2837 | . . 3 ⊢ (𝑥 = 𝑦 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝑦) = 𝑦)) |
7 | oveq2 7164 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1 · 𝑥) = (1 · (𝑦 + 1))) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → 𝑥 = (𝑦 + 1)) | |
9 | 7, 8 | eqeq12d 2837 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((1 · 𝑥) = 𝑥 ↔ (1 · (𝑦 + 1)) = (𝑦 + 1))) |
10 | oveq2 7164 | . . . 4 ⊢ (𝑥 = 𝐴 → (1 · 𝑥) = (1 · 𝐴)) | |
11 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2837 | . . 3 ⊢ (𝑥 = 𝐴 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝐴) = 𝐴)) |
13 | 1t1e1ALT 39204 | . . 3 ⊢ (1 · 1) = 1 | |
14 | 1cnd 10636 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 1 ∈ ℂ) | |
15 | simpl 485 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℕ) | |
16 | 15 | nncnd 11654 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℂ) |
17 | 14, 16, 14 | adddid 10665 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = ((1 · 𝑦) + (1 · 1))) |
18 | simpr 487 | . . . . . 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 7174 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → ((1 · 𝑦) + (1 · 1)) = (𝑦 + 1)) |
21 | 17, 20 | eqtrd 2856 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → (1 · (𝑦 + 1)) = (𝑦 + 1)) |
22 | 21 | ex 415 | . . 3 ⊢ (𝑦 ∈ ℕ → ((1 · 𝑦) = 𝑦 → (1 · (𝑦 + 1)) = (𝑦 + 1))) |
23 | 3, 6, 9, 12, 13, 22 | nnind 11656 | . 2 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = 𝐴) |
24 | nnre 11645 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
25 | ax-1rid 10607 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 23, 26 | eqtr4d 2859 | 1 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rrecex 10609 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 |
This theorem is referenced by: nnmulcom 39214 |
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