MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mumul Structured version   Visualization version   GIF version

Theorem mumul 27106
Description: The MΓΆbius function is a multiplicative function. This is one of the primary interests of the MΓΆbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
mumul ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))

Proof of Theorem mumul
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1190 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ 𝐡 ∈ β„•)
2 mucl 27066 . . . . . 6 (𝐡 ∈ β„• β†’ (ΞΌβ€˜π΅) ∈ β„€)
31, 2syl 17 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΅) ∈ β„€)
43zcnd 12691 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΅) ∈ β„‚)
54mul02d 11436 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (0 Β· (ΞΌβ€˜π΅)) = 0)
6 simpr 484 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΄) = 0)
76oveq1d 7429 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = (0 Β· (ΞΌβ€˜π΅)))
8 mumullem1 27104 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
983adantl3 1166 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
105, 7, 93eqtr4rd 2779 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
11 simpl1 1189 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ 𝐴 ∈ β„•)
12 mucl 27066 . . . . . 6 (𝐴 ∈ β„• β†’ (ΞΌβ€˜π΄) ∈ β„€)
1311, 12syl 17 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΄) ∈ β„€)
1413zcnd 12691 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΄) ∈ β„‚)
1514mul01d 11437 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ ((ΞΌβ€˜π΄) Β· 0) = 0)
16 simpr 484 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΅) = 0)
1716oveq2d 7430 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = ((ΞΌβ€˜π΄) Β· 0))
18 nncn 12244 . . . . . . . 8 (𝐴 ∈ β„• β†’ 𝐴 ∈ β„‚)
19 nncn 12244 . . . . . . . 8 (𝐡 ∈ β„• β†’ 𝐡 ∈ β„‚)
20 mulcom 11218 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
2118, 19, 20syl2an 595 . . . . . . 7 ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
2221fveq2d 6895 . . . . . 6 ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (ΞΌβ€˜(𝐡 Β· 𝐴)))
2322adantr 480 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (ΞΌβ€˜(𝐡 Β· 𝐴)))
24 mumullem1 27104 . . . . . 6 (((𝐡 ∈ β„• ∧ 𝐴 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐡 Β· 𝐴)) = 0)
2524ancom1s 652 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐡 Β· 𝐴)) = 0)
2623, 25eqtrd 2768 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
27263adantl3 1166 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
2815, 17, 273eqtr4rd 2779 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
29 simpl1 1189 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐴 ∈ β„•)
30 simpl2 1190 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐡 ∈ β„•)
3129, 30nnmulcld 12289 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (𝐴 Β· 𝐡) ∈ β„•)
32 mumullem2 27105 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) β‰  0)
33 muval2 27059 . . . 4 (((𝐴 Β· 𝐡) ∈ β„• ∧ (ΞΌβ€˜(𝐴 Β· 𝐡)) β‰  0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)})))
3431, 32, 33syl2anc 583 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)})))
35 neg1cn 12350 . . . . . 6 -1 ∈ β„‚
3635a1i 11 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ -1 ∈ β„‚)
37 fzfi 13963 . . . . . . 7 (1...𝐡) ∈ Fin
38 prmssnn 16640 . . . . . . . . 9 β„™ βŠ† β„•
39 rabss2 4071 . . . . . . . . 9 (β„™ βŠ† β„• β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} βŠ† {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐡})
4038, 39ax-mp 5 . . . . . . . 8 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} βŠ† {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐡}
41 dvdsssfz1 16288 . . . . . . . . 9 (𝐡 ∈ β„• β†’ {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐡} βŠ† (1...𝐡))
4230, 41syl 17 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐡} βŠ† (1...𝐡))
4340, 42sstrid 3989 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} βŠ† (1...𝐡))
44 ssfi 9191 . . . . . . 7 (((1...𝐡) ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} βŠ† (1...𝐡)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} ∈ Fin)
4537, 43, 44sylancr 586 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} ∈ Fin)
46 hashcl 14341 . . . . . 6 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} ∈ Fin β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) ∈ β„•0)
4745, 46syl 17 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) ∈ β„•0)
48 fzfi 13963 . . . . . . 7 (1...𝐴) ∈ Fin
49 rabss2 4071 . . . . . . . . 9 (β„™ βŠ† β„• β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βŠ† {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐴})
5038, 49ax-mp 5 . . . . . . . 8 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βŠ† {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐴}
51 dvdsssfz1 16288 . . . . . . . . 9 (𝐴 ∈ β„• β†’ {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐴} βŠ† (1...𝐴))
5229, 51syl 17 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„• ∣ 𝑝 βˆ₯ 𝐴} βŠ† (1...𝐴))
5350, 52sstrid 3989 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βŠ† (1...𝐴))
54 ssfi 9191 . . . . . . 7 (((1...𝐴) ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βŠ† (1...𝐴)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∈ Fin)
5548, 53, 54sylancr 586 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∈ Fin)
56 hashcl 14341 . . . . . 6 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∈ Fin β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) ∈ β„•0)
5755, 56syl 17 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) ∈ β„•0)
5836, 47, 57expaddd 14138 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (-1↑((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) + (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}))) = ((-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})) Β· (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}))))
59 simpr 484 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝑝 ∈ β„™)
60 simpl1 1189 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„•)
6160nnzd 12609 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„€)
6261adantlr 714 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„€)
63 simpl2 1190 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„•)
6463nnzd 12609 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„€)
6564adantlr 714 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„€)
66 euclemma 16677 . . . . . . . . . 10 ((𝑝 ∈ β„™ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝑝 βˆ₯ (𝐴 Β· 𝐡) ↔ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)))
6759, 62, 65, 66syl3anc 1369 . . . . . . . . 9 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ (𝑝 βˆ₯ (𝐴 Β· 𝐡) ↔ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)))
6867rabbidva 3435 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)} = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)})
69 unrab 4301 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)}
7068, 69eqtr4di 2786 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)} = ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}))
7170fveq2d 6895 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)}) = (β™―β€˜({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
72 inrab 4302 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∩ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡)}
73 nprmdvds1 16670 . . . . . . . . . . . 12 (𝑝 ∈ β„™ β†’ Β¬ 𝑝 βˆ₯ 1)
7473adantl 481 . . . . . . . . . . 11 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ Β¬ 𝑝 βˆ₯ 1)
75 prmz 16639 . . . . . . . . . . . . . 14 (𝑝 ∈ β„™ β†’ 𝑝 ∈ β„€)
7675adantl 481 . . . . . . . . . . . . 13 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝑝 ∈ β„€)
77 dvdsgcd 16513 . . . . . . . . . . . . 13 ((𝑝 ∈ β„€ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ ((𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡) β†’ 𝑝 βˆ₯ (𝐴 gcd 𝐡)))
7876, 62, 65, 77syl3anc 1369 . . . . . . . . . . . 12 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ ((𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡) β†’ 𝑝 βˆ₯ (𝐴 gcd 𝐡)))
79 simpll3 1212 . . . . . . . . . . . . 13 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ (𝐴 gcd 𝐡) = 1)
8079breq2d 5154 . . . . . . . . . . . 12 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ (𝑝 βˆ₯ (𝐴 gcd 𝐡) ↔ 𝑝 βˆ₯ 1))
8178, 80sylibd 238 . . . . . . . . . . 11 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ ((𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡) β†’ 𝑝 βˆ₯ 1))
8274, 81mtod 197 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ Β¬ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡))
8382ralrimiva 3142 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ βˆ€π‘ ∈ β„™ Β¬ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡))
84 rabeq0 4380 . . . . . . . . 9 ({𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡)} = βˆ… ↔ βˆ€π‘ ∈ β„™ Β¬ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡))
8583, 84sylibr 233 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡)} = βˆ…)
8672, 85eqtrid 2780 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∩ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = βˆ…)
87 hashun 14367 . . . . . . 7 (({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} ∈ Fin ∧ ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∩ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = βˆ…) β†’ (β™―β€˜({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})) = ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) + (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
8855, 45, 86, 87syl3anc 1369 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (β™―β€˜({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})) = ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) + (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
8971, 88eqtrd 2768 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)}) = ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) + (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
9089oveq2d 7430 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)})) = (-1↑((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}) + (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}))))
91 simprl 770 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΄) β‰  0)
92 muval2 27059 . . . . . 6 ((𝐴 ∈ β„• ∧ (ΞΌβ€˜π΄) β‰  0) β†’ (ΞΌβ€˜π΄) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})))
9329, 91, 92syl2anc 583 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΄) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})))
94 simprr 772 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΅) β‰  0)
95 muval2 27059 . . . . . 6 ((𝐡 ∈ β„• ∧ (ΞΌβ€˜π΅) β‰  0) β†’ (ΞΌβ€˜π΅) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
9630, 94, 95syl2anc 583 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΅) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
9793, 96oveq12d 7432 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = ((-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})) Β· (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}))))
9858, 90, 973eqtr4rd 2779 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ (𝐴 Β· 𝐡)})))
9934, 98eqtr4d 2771 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
10010, 28, 99pm2.61da2ne 3026 1 ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2936  βˆ€wral 3057  {crab 3428   βˆͺ cun 3943   ∩ cin 3944   βŠ† wss 3945  βˆ…c0 4318   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Fincfn 8957  β„‚cc 11130  0cc0 11132  1c1 11133   + caddc 11135   Β· cmul 11137  -cneg 11469  β„•cn 12236  β„•0cn0 12496  β„€cz 12582  ...cfz 13510  β†‘cexp 14052  β™―chash 14315   βˆ₯ cdvds 16224   gcd cgcd 16462  β„™cprime 16635  ΞΌcmu 27020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-inf 9460  df-dju 9918  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-fz 13511  df-fl 13783  df-mod 13861  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16225  df-gcd 16463  df-prm 16636  df-pc 16799  df-mu 27026
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator