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Mirrors > Home > MPE Home > Th. List > Mathboxes > gcdcomnni | Structured version Visualization version GIF version |
Description: Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
gcdcomnni.1 | ⊢ 𝑀 ∈ ℕ |
gcdcomnni.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
gcdcomnni | ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdcomnni.1 | . . . 4 ⊢ 𝑀 ∈ ℕ | |
2 | 1 | nnzi 11993 | . . 3 ⊢ 𝑀 ∈ ℤ |
3 | gcdcomnni.2 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
4 | 3 | nnzi 11993 | . . 3 ⊢ 𝑁 ∈ ℤ |
5 | 2, 4 | pm3.2i 473 | . 2 ⊢ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) |
6 | gcdcom 15845 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7142 ℕcn 11624 ℤcz 11968 gcd cgcd 15826 |
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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-i2m1 10591 ax-1ne0 10592 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 |
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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-pnf 10663 df-mnf 10664 df-ltxr 10666 df-neg 10859 df-nn 11625 df-z 11969 df-gcd 15827 |
This theorem is referenced by: 12gcd5e1 39141 60gcd6e6 39142 60gcd7e1 39143 420gcd8e4 39144 |
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