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 3823 | . . 3 ⊢ (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) | |
2 | nzprmdif.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℙ) | |
3 | prmz 15227 | . . . . . 6 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | nzprmdif.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℙ) | |
6 | prmz 15227 | . . . . . 6 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 4, 7 | nzin 37539 | . . . 4 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) |
9 | 8 | difeq2d 3690 | . . 3 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
10 | 1, 9 | syl5eqr 2658 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
11 | lcmgcd 15158 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
12 | 4, 7, 11 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
13 | nzprmdif.ne | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
14 | prmrp 15262 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℙ ∧ 𝑁 ∈ ℙ) → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) | |
15 | 2, 5, 14 | syl2anc 691 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) |
16 | 13, 15 | mpbird 246 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
17 | 16 | oveq2d 6565 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 1)) |
18 | lcmcl 15152 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
19 | 4, 7, 18 | syl2anc 691 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0) |
20 | 19 | nn0cnd 11230 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℂ) |
21 | 20 | mulid1d 9936 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · 1) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | eqtrd 2644 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
23 | 4 | zred 11358 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 7 | zred 11358 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 23, 24 | remulcld 9949 | . . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℝ) |
26 | prmnn 15226 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℕ) | |
27 | 2, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | 27 | nnnn0d 11228 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
29 | 28 | nn0ge0d 11231 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑀) |
30 | prmnn 15226 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
32 | 31 | nnnn0d 11228 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 32 | nn0ge0d 11231 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑁) |
34 | 23, 24, 29, 33 | mulge0d 10483 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑁)) |
35 | 25, 34 | absidd 14009 | . . . . . 6 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
36 | 12, 22, 35 | 3eqtr3d 2652 | . . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) = (𝑀 · 𝑁)) |
37 | 36 | sneqd 4137 | . . . 4 ⊢ (𝜑 → {(𝑀 lcm 𝑁)} = {(𝑀 · 𝑁)}) |
38 | 37 | imaeq2d 5385 | . . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) = ( ∥ “ {(𝑀 · 𝑁)})) |
39 | 38 | difeq2d 3690 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
40 | 10, 39 | eqtrd 2644 | 1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∩ cin 3539 {csn 4125 “ cima 5041 ‘cfv 5804 (class class class)co 6549 1c1 9816 · cmul 9820 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 abscabs 13822 ∥ cdvds 14821 gcd cgcd 15054 lcm clcm 15139 ℙcprime 15223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-lcm 15141 df-prm 15224 |
This theorem is referenced by: (None) |
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