Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
||
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 4238 | . . 3 ⊢ (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) | |
2 | nzprmdif.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℙ) | |
3 | prmz 16019 | . . . . . 6 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | nzprmdif.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℙ) | |
6 | prmz 16019 | . . . . . 6 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 4, 7 | nzin 40699 | . . . 4 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) |
9 | 8 | difeq2d 4099 | . . 3 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
10 | 1, 9 | syl5eqr 2870 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
11 | lcmgcd 15951 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
12 | 4, 7, 11 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
13 | nzprmdif.ne | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
14 | prmrp 16056 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℙ ∧ 𝑁 ∈ ℙ) → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) | |
15 | 2, 5, 14 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) |
16 | 13, 15 | mpbird 259 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
17 | 16 | oveq2d 7172 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 1)) |
18 | lcmcl 15945 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
19 | 4, 7, 18 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0) |
20 | 19 | nn0cnd 11958 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℂ) |
21 | 20 | mulid1d 10658 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · 1) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | eqtrd 2856 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
23 | 4 | zred 12088 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 7 | zred 12088 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 23, 24 | remulcld 10671 | . . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℝ) |
26 | prmnn 16018 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℕ) | |
27 | 2, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | 27 | nnnn0d 11956 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
29 | 28 | nn0ge0d 11959 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑀) |
30 | prmnn 16018 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
32 | 31 | nnnn0d 11956 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 32 | nn0ge0d 11959 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑁) |
34 | 23, 24, 29, 33 | mulge0d 11217 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑁)) |
35 | 25, 34 | absidd 14782 | . . . . . 6 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
36 | 12, 22, 35 | 3eqtr3d 2864 | . . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) = (𝑀 · 𝑁)) |
37 | 36 | sneqd 4579 | . . . 4 ⊢ (𝜑 → {(𝑀 lcm 𝑁)} = {(𝑀 · 𝑁)}) |
38 | 37 | imaeq2d 5929 | . . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) = ( ∥ “ {(𝑀 · 𝑁)})) |
39 | 38 | difeq2d 4099 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
40 | 10, 39 | eqtrd 2856 | 1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ∩ cin 3935 {csn 4567 “ cima 5558 ‘cfv 6355 (class class class)co 7156 1c1 10538 · cmul 10542 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 abscabs 14593 ∥ cdvds 15607 gcd cgcd 15843 lcm clcm 15932 ℙcprime 16015 |
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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-lcm 15934 df-prm 16016 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |