![]() |
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 4119 | . . 3 ⊢ (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) | |
2 | nzprmdif.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℙ) | |
3 | prmz 15873 | . . . . . 6 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | nzprmdif.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℙ) | |
6 | prmz 15873 | . . . . . 6 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 4, 7 | nzin 40095 | . . . 4 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) |
9 | 8 | difeq2d 3983 | . . 3 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁}))) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
10 | 1, 9 | syl5eqr 2822 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)}))) |
11 | lcmgcd 15805 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
12 | 4, 7, 11 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
13 | nzprmdif.ne | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
14 | prmrp 15910 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℙ ∧ 𝑁 ∈ ℙ) → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) | |
15 | 2, 5, 14 | syl2anc 576 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 gcd 𝑁) = 1 ↔ 𝑀 ≠ 𝑁)) |
16 | 13, 15 | mpbird 249 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
17 | 16 | oveq2d 6990 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 1)) |
18 | lcmcl 15799 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
19 | 4, 7, 18 | syl2anc 576 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0) |
20 | 19 | nn0cnd 11767 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℂ) |
21 | 20 | mulid1d 10455 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · 1) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | eqtrd 2808 | . . . . . 6 ⊢ (𝜑 → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
23 | 4 | zred 11898 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 7 | zred 11898 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 23, 24 | remulcld 10468 | . . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℝ) |
26 | prmnn 15872 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℙ → 𝑀 ∈ ℕ) | |
27 | 2, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | 27 | nnnn0d 11765 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
29 | 28 | nn0ge0d 11768 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑀) |
30 | prmnn 15872 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
32 | 31 | nnnn0d 11765 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 32 | nn0ge0d 11768 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑁) |
34 | 23, 24, 29, 33 | mulge0d 11016 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑁)) |
35 | 25, 34 | absidd 14641 | . . . . . 6 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
36 | 12, 22, 35 | 3eqtr3d 2816 | . . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) = (𝑀 · 𝑁)) |
37 | 36 | sneqd 4447 | . . . 4 ⊢ (𝜑 → {(𝑀 lcm 𝑁)} = {(𝑀 · 𝑁)}) |
38 | 37 | imaeq2d 5767 | . . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) = ( ∥ “ {(𝑀 · 𝑁)})) |
39 | 38 | difeq2d 3983 | . 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 lcm 𝑁)})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
40 | 10, 39 | eqtrd 2808 | 1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∖ cdif 3820 ∩ cin 3822 {csn 4435 “ cima 5406 ‘cfv 6185 (class class class)co 6974 1c1 10334 · cmul 10338 ℕcn 11437 ℕ0cn0 11705 ℤcz 11791 abscabs 14452 ∥ cdvds 15465 gcd cgcd 15701 lcm clcm 15786 ℙcprime 15869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-fl 12975 df-mod 13051 df-seq 13183 df-exp 13243 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-dvds 15466 df-gcd 15702 df-lcm 15788 df-prm 15870 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |