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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulassnni | Structured version Visualization version GIF version |
Description: Associative law for multiplication. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
mulassnni.1 | ⊢ 𝐴 ∈ ℕ |
mulassnni.2 | ⊢ 𝐵 ∈ ℕ |
mulassnni.3 | ⊢ 𝐶 ∈ ℕ |
Ref | Expression |
---|---|
mulassnni | ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulassnni.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
2 | 1 | nncni 11623 | . 2 ⊢ 𝐴 ∈ ℂ |
3 | mulassnni.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
4 | 3 | nncni 11623 | . 2 ⊢ 𝐵 ∈ ℂ |
5 | mulassnni.3 | . . 3 ⊢ 𝐶 ∈ ℕ | |
6 | 5 | nncni 11623 | . 2 ⊢ 𝐶 ∈ ℂ |
7 | 2, 4, 6 | mulassi 10627 | 1 ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7131 · cmul 10517 ℕcn 11613 |
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 2792 ax-sep 5177 ax-nul 5184 ax-pow 5240 ax-pr 5304 ax-un 7436 ax-1cn 10570 ax-addcl 10572 ax-mulass 10578 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3007 df-ral 3130 df-rex 3131 df-reu 3132 df-rab 3134 df-v 3475 df-sbc 3752 df-csb 3860 df-dif 3915 df-un 3917 df-in 3919 df-ss 3928 df-pss 3930 df-nul 4268 df-if 4442 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4813 df-iun 4895 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5434 df-eprel 5439 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6122 df-ord 6168 df-on 6169 df-lim 6170 df-suc 6171 df-iota 6288 df-fun 6331 df-fn 6332 df-f 6333 df-f1 6334 df-fo 6335 df-f1o 6336 df-fv 6337 df-ov 7134 df-om 7556 df-wrecs 7922 df-recs 7983 df-rdg 8021 df-nn 11614 |
This theorem is referenced by: 420lcm8e840 39143 |
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