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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 4262 | . . 3 ⊢ (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) | |
2 | nzprmdif.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℙ) | |
3 | prmz 16671 | . . . . . 6 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | nzprmdif.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℙ) | |
6 | prmz 16671 | . . . . . 6 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 4, 7 | nzin 43929 | . . . 4 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) |
9 | 8 | difeq2d 4120 | . . 3 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
10 | 1, 9 | eqtr3id 2779 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
11 | lcmgcd 16603 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
12 | 4, 7, 11 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
13 | nzprmdif.ne | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
14 | prmrp 16708 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℙ ∧ 𝑁 ∈ ℙ) → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) | |
15 | 2, 5, 14 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) |
16 | 13, 15 | mpbird 256 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
17 | 16 | oveq2d 7439 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 1)) |
18 | lcmcl 16597 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
19 | 4, 7, 18 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0) |
20 | 19 | nn0cnd 12581 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℂ) |
21 | 20 | mulridd 11277 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · 1) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | eqtrd 2765 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
23 | 4 | zred 12713 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 7 | zred 12713 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 23, 24 | remulcld 11290 | . . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℝ) |
26 | prmnn 16670 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℕ) | |
27 | 2, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | 27 | nnnn0d 12579 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
29 | 28 | nn0ge0d 12582 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑀) |
30 | prmnn 16670 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
32 | 31 | nnnn0d 12579 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 32 | nn0ge0d 12582 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑁) |
34 | 23, 24, 29, 33 | mulge0d 11837 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑁)) |
35 | 25, 34 | absidd 15422 | . . . . . 6 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
36 | 12, 22, 35 | 3eqtr3d 2773 | . . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) = (𝑀 · 𝑁)) |
37 | 36 | sneqd 4644 | . . . 4 ⊢ (𝜑 → {(𝑀 lcm 𝑁)} = {(𝑀 · 𝑁)}) |
38 | 37 | imaeq2d 6068 | . . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) = ( ∥ “ {(𝑀 · 𝑁)})) |
39 | 38 | difeq2d 4120 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
40 | 10, 39 | eqtrd 2765 | 1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3943 ∩ cin 3945 {csn 4632 “ cima 5684 ‘cfv 6553 (class class class)co 7423 1c1 11155 · cmul 11159 ℕcn 12259 ℕ0cn0 12519 ℤcz 12605 abscabs 15234 ∥ cdvds 16251 gcd cgcd 16489 lcm clcm 16584 ℙcprime 16667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-fl 13807 df-mod 13885 df-seq 14017 df-exp 14077 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-gcd 16490 df-lcm 16586 df-prm 16668 |
This theorem is referenced by: (None) |
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