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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 10594. Since (𝐴 · 1) is 𝐴 by ax-1rid 10600, this is equivalent to remulid2 39332 for natural numbers, but using fewer axioms (avoiding ax-resscn 10587, ax-addass 10595, ax-mulass 10596, ax-rnegex 10601, ax-pre-lttri 10604, ax-pre-lttrn 10605, ax-pre-ltadd 10606). (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
nnmul1com | ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7157 | . . . 4 ⊢ (𝑥 = 1 → (1 · 𝑥) = (1 · 1)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
3 | 1, 2 | eqeq12d 2836 | . . 3 ⊢ (𝑥 = 1 → ((1 · 𝑥) = 𝑥 ↔ (1 · 1) = 1)) |
4 | oveq2 7157 | . . . 4 ⊢ (𝑥 = 𝑦 → (1 · 𝑥) = (1 · 𝑦)) | |
5 | id 22 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2836 | . . 3 ⊢ (𝑥 = 𝑦 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝑦) = 𝑦)) |
7 | oveq2 7157 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1 · 𝑥) = (1 · (𝑦 + 1))) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → 𝑥 = (𝑦 + 1)) | |
9 | 7, 8 | eqeq12d 2836 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((1 · 𝑥) = 𝑥 ↔ (1 · (𝑦 + 1)) = (𝑦 + 1))) |
10 | oveq2 7157 | . . . 4 ⊢ (𝑥 = 𝐴 → (1 · 𝑥) = (1 · 𝐴)) | |
11 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2836 | . . 3 ⊢ (𝑥 = 𝐴 → ((1 · 𝑥) = 𝑥 ↔ (1 · 𝐴) = 𝐴)) |
13 | 1t1e1ALT 39238 | . . 3 ⊢ (1 · 1) = 1 | |
14 | 1cnd 10629 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 1 ∈ ℂ) | |
15 | simpl 485 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℕ) | |
16 | 15 | nncnd 11647 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → 𝑦 ∈ ℂ) |
17 | 14, 16, 14 | adddid 10658 | . . . . 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 7167 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ (1 · 𝑦) = 𝑦) → ((1 · 𝑦) + (1 · 1)) = (𝑦 + 1)) |
21 | 17, 20 | eqtrd 2855 | . . . 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 11649 | . 2 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = 𝐴) |
24 | nnre 11638 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
25 | ax-1rid 10600 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 23, 26 | eqtr4d 2858 | 1 ⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ℝcr 10529 1c1 10531 + caddc 10533 · cmul 10535 ℕcn 11631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rrecex 10602 ax-cnre 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-nn 11632 |
This theorem is referenced by: nnmulcom 39248 |
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