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

Theorem mumul 26553
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 1193 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ 𝐡 ∈ β„•)
2 mucl 26513 . . . . . 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 12616 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΅) ∈ β„‚)
54mul02d 11361 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (0 Β· (ΞΌβ€˜π΅)) = 0)
6 simpr 486 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΄) = 0)
76oveq1d 7376 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = (0 Β· (ΞΌβ€˜π΅)))
8 mumullem1 26551 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
983adantl3 1169 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
105, 7, 93eqtr4rd 2784 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
11 simpl1 1192 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ 𝐴 ∈ β„•)
12 mucl 26513 . . . . . 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 12616 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΄) ∈ β„‚)
1514mul01d 11362 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ ((ΞΌβ€˜π΄) Β· 0) = 0)
16 simpr 486 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΅) = 0)
1716oveq2d 7377 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = ((ΞΌβ€˜π΄) Β· 0))
18 nncn 12169 . . . . . . . 8 (𝐴 ∈ β„• β†’ 𝐴 ∈ β„‚)
19 nncn 12169 . . . . . . . 8 (𝐡 ∈ β„• β†’ 𝐡 ∈ β„‚)
20 mulcom 11145 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
2118, 19, 20syl2an 597 . . . . . . 7 ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
2221fveq2d 6850 . . . . . 6 ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (ΞΌβ€˜(𝐡 Β· 𝐴)))
2322adantr 482 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = (ΞΌβ€˜(𝐡 Β· 𝐴)))
24 mumullem1 26551 . . . . . 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 2773 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
27263adantl3 1169 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
2815, 17, 273eqtr4rd 2784 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
29 simpl1 1192 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐴 ∈ β„•)
30 simpl2 1193 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐡 ∈ β„•)
3129, 30nnmulcld 12214 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (𝐴 Β· 𝐡) ∈ β„•)
32 mumullem2 26552 . . . 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 26506 . . . 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 585 . . 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 12275 . . . . . 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 13886 . . . . . . 7 (1...𝐡) ∈ Fin
38 prmssnn 16560 . . . . . . . . 9 β„™ βŠ† β„•
39 rabss2 4039 . . . . . . . . 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 16208 . . . . . . . . 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 3959 . . . . . . 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 9123 . . . . . . 7 (((1...𝐡) ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} βŠ† (1...𝐡)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡} ∈ Fin)
4537, 43, 44sylancr 588 . . . . . 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 14265 . . . . . 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 13886 . . . . . . 7 (1...𝐴) ∈ Fin
49 rabss2 4039 . . . . . . . . 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 16208 . . . . . . . . 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 3959 . . . . . . 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 9123 . . . . . . 7 (((1...𝐴) ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βŠ† (1...𝐴)) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∈ Fin)
5548, 53, 54sylancr 588 . . . . . 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 14265 . . . . . 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 14062 . . . 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 486 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝑝 ∈ β„™)
60 simpl1 1192 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„•)
6160nnzd 12534 . . . . . . . . . . 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 1193 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„•)
6463nnzd 12534 . . . . . . . . . . 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 16597 . . . . . . . . . 10 ((𝑝 ∈ β„™ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝑝 βˆ₯ (𝐴 Β· 𝐡) ↔ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)))
6759, 62, 65, 66syl3anc 1372 . . . . . . . . 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 3413 . . . . . . . 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 4269 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)}
7068, 69eqtr4di 2791 . . . . . . 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 6850 . . . . . 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 4270 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∩ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡)}
73 nprmdvds1 16590 . . . . . . . . . . . 12 (𝑝 ∈ β„™ β†’ Β¬ 𝑝 βˆ₯ 1)
7473adantl 483 . . . . . . . . . . 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 16559 . . . . . . . . . . . . . 14 (𝑝 ∈ β„™ β†’ 𝑝 ∈ β„€)
7675adantl 483 . . . . . . . . . . . . 13 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝑝 ∈ β„€)
77 dvdsgcd 16433 . . . . . . . . . . . . 13 ((𝑝 ∈ β„€ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ ((𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡) β†’ 𝑝 βˆ₯ (𝐴 gcd 𝐡)))
7876, 62, 65, 77syl3anc 1372 . . . . . . . . . . . 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 1215 . . . . . . . . . . . . 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 5121 . . . . . . . . . . . 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 3140 . . . . . . . . 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 4348 . . . . . . . . 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 2785 . . . . . . 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 14291 . . . . . . 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 1372 . . . . . 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 2773 . . . . 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 7377 . . . 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 26506 . . . . . 6 ((𝐴 ∈ β„• ∧ (ΞΌβ€˜π΄) β‰  0) β†’ (ΞΌβ€˜π΄) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})))
9329, 91, 92syl2anc 585 . . . . 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 26506 . . . . . 6 ((𝐡 ∈ β„• ∧ (ΞΌβ€˜π΅) β‰  0) β†’ (ΞΌβ€˜π΅) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡})))
9630, 94, 95syl2anc 585 . . . . 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 7379 . . . 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 2784 . . 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 2776 . 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 3030 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889  β„‚cc 11057  0cc0 11059  1c1 11060   + caddc 11062   Β· cmul 11064  -cneg 11394  β„•cn 12161  β„•0cn0 12421  β„€cz 12507  ...cfz 13433  β†‘cexp 13976  β™―chash 14239   βˆ₯ cdvds 16144   gcd cgcd 16382  β„™cprime 16555  ΞΌcmu 26467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-fz 13434  df-fl 13706  df-mod 13784  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-dvds 16145  df-gcd 16383  df-prm 16556  df-pc 16717  df-mu 26473
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator