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Mirrors > Home > MPE Home > Th. List > lcmfpr | Structured version Visualization version GIF version |
Description: The value of the lcm function for an unordered pair is the value of the lcm operator for both elements. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
Ref | Expression |
---|---|
lcmfpr | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (lcm‘{𝑀, 𝑁}) = (𝑀 lcm 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11253 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | elpr 4655 | . . . . 5 ⊢ (0 ∈ {𝑀, 𝑁} ↔ (0 = 𝑀 ∨ 0 = 𝑁)) |
3 | eqcom 2742 | . . . . . 6 ⊢ (0 = 𝑀 ↔ 𝑀 = 0) | |
4 | eqcom 2742 | . . . . . 6 ⊢ (0 = 𝑁 ↔ 𝑁 = 0) | |
5 | 3, 4 | orbi12i 914 | . . . . 5 ⊢ ((0 = 𝑀 ∨ 0 = 𝑁) ↔ (𝑀 = 0 ∨ 𝑁 = 0)) |
6 | 2, 5 | bitri 275 | . . . 4 ⊢ (0 ∈ {𝑀, 𝑁} ↔ (𝑀 = 0 ∨ 𝑁 = 0)) |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∈ {𝑀, 𝑁} ↔ (𝑀 = 0 ∨ 𝑁 = 0))) |
8 | breq1 5151 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 ∥ 𝑛 ↔ 𝑀 ∥ 𝑛)) | |
9 | breq1 5151 | . . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 ∥ 𝑛 ↔ 𝑁 ∥ 𝑛)) | |
10 | 8, 9 | ralprg 4701 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛 ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
11 | 10 | rabbidv 3441 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
12 | 11 | infeq1d 9515 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) |
13 | 7, 12 | ifbieq2d 4557 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if(0 ∈ {𝑀, 𝑁}, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
14 | prssi 4826 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑀, 𝑁} ⊆ ℤ) | |
15 | prfi 9361 | . . 3 ⊢ {𝑀, 𝑁} ∈ Fin | |
16 | lcmfval 16655 | . . 3 ⊢ (({𝑀, 𝑁} ⊆ ℤ ∧ {𝑀, 𝑁} ∈ Fin) → (lcm‘{𝑀, 𝑁}) = if(0 ∈ {𝑀, 𝑁}, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛}, ℝ, < ))) | |
17 | 14, 15, 16 | sylancl 586 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (lcm‘{𝑀, 𝑁}) = if(0 ∈ {𝑀, 𝑁}, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ {𝑀, 𝑁}𝑚 ∥ 𝑛}, ℝ, < ))) |
18 | lcmval 16626 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) | |
19 | 13, 17, 18 | 3eqtr4d 2785 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (lcm‘{𝑀, 𝑁}) = (𝑀 lcm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 ifcif 4531 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 infcinf 9479 ℝcr 11152 0cc0 11153 < clt 11293 ℕcn 12264 ℤcz 12611 ∥ cdvds 16287 lcm clcm 16622 lcmclcmf 16623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-dvds 16288 df-lcm 16624 df-lcmf 16625 |
This theorem is referenced by: lcmfsn 16669 |
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