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Mathbox for Steve Rodriguez |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nzprmdif | Structured version Visualization version GIF version |
Description: Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
nzprmdif.m | ⊢ (𝜑 → 𝑀 ∈ ℙ) |
nzprmdif.n | ⊢ (𝜑 → 𝑁 ∈ ℙ) |
nzprmdif.ne | ⊢ (𝜑 → 𝑀 ≠ 𝑁) |
Ref | Expression |
---|---|
nzprmdif | ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difin 4188 | . . 3 ⊢ (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) | |
2 | nzprmdif.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℙ) | |
3 | prmz 16009 | . . . . . 6 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | nzprmdif.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℙ) | |
6 | prmz 16009 | . . . . . 6 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 4, 7 | nzin 41022 | . . . 4 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) |
9 | 8 | difeq2d 4050 | . . 3 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
10 | 1, 9 | syl5eqr 2847 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
11 | lcmgcd 15941 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
12 | 4, 7, 11 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
13 | nzprmdif.ne | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
14 | prmrp 16046 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℙ ∧ 𝑁 ∈ ℙ) → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) | |
15 | 2, 5, 14 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) |
16 | 13, 15 | mpbird 260 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
17 | 16 | oveq2d 7151 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 1)) |
18 | lcmcl 15935 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
19 | 4, 7, 18 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0) |
20 | 19 | nn0cnd 11945 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℂ) |
21 | 20 | mulid1d 10647 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · 1) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | eqtrd 2833 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
23 | 4 | zred 12075 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 7 | zred 12075 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 23, 24 | remulcld 10660 | . . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℝ) |
26 | prmnn 16008 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℕ) | |
27 | 2, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | 27 | nnnn0d 11943 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
29 | 28 | nn0ge0d 11946 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑀) |
30 | prmnn 16008 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
32 | 31 | nnnn0d 11943 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 32 | nn0ge0d 11946 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑁) |
34 | 23, 24, 29, 33 | mulge0d 11206 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑁)) |
35 | 25, 34 | absidd 14774 | . . . . . 6 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
36 | 12, 22, 35 | 3eqtr3d 2841 | . . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) = (𝑀 · 𝑁)) |
37 | 36 | sneqd 4537 | . . . 4 ⊢ (𝜑 → {(𝑀 lcm 𝑁)} = {(𝑀 · 𝑁)}) |
38 | 37 | imaeq2d 5896 | . . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) = ( ∥ “ {(𝑀 · 𝑁)})) |
39 | 38 | difeq2d 4050 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
40 | 10, 39 | eqtrd 2833 | 1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ∩ cin 3880 {csn 4525 “ cima 5522 ‘cfv 6324 (class class class)co 7135 1c1 10527 · cmul 10531 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 abscabs 14585 ∥ cdvds 15599 gcd cgcd 15833 lcm clcm 15922 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-lcm 15924 df-prm 16006 |
This theorem is referenced by: (None) |
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