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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulcomnni | Structured version Visualization version GIF version |
Description: Commutative law for multiplication. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
mulcomnni.1 | ⊢ 𝐴 ∈ ℕ |
mulcomnni.2 | ⊢ 𝐵 ∈ ℕ |
Ref | Expression |
---|---|
mulcomnni | ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcomnni.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
2 | 1 | nncni 12227 | . 2 ⊢ 𝐴 ∈ ℂ |
3 | mulcomnni.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
4 | 3 | nncni 12227 | . 2 ⊢ 𝐵 ∈ ℂ |
5 | 2, 4 | mulcomi 11227 | 1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 (class class class)co 7412 · cmul 11119 ℕcn 12217 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-1cn 11172 ax-addcl 11174 ax-mulcom 11178 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12218 |
This theorem is referenced by: 12gcd5e1 41175 60gcd6e6 41176 420gcd8e4 41178 lcmeprodgcdi 41179 60lcm6e60 41181 420lcm8e840 41183 |
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