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Theorem mumul 27032
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 1189 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ 𝐡 ∈ β„•)
2 mucl 26992 . . . . . 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 12665 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜π΅) ∈ β„‚)
54mul02d 11410 . . 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 7417 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = (0 Β· (ΞΌβ€˜π΅)))
8 mumullem1 27030 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
983adantl3 1165 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
105, 7, 93eqtr4rd 2775 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΄) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
11 simpl1 1188 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ 𝐴 ∈ β„•)
12 mucl 26992 . . . . . 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 12665 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜π΄) ∈ β„‚)
1514mul01d 11411 . . 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 7418 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)) = ((ΞΌβ€˜π΄) Β· 0))
18 nncn 12218 . . . . . . . 8 (𝐴 ∈ β„• β†’ 𝐴 ∈ β„‚)
19 nncn 12218 . . . . . . . 8 (𝐡 ∈ β„• β†’ 𝐡 ∈ β„‚)
20 mulcom 11193 . . . . . . . 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 6886 . . . . . 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 27030 . . . . . 6 (((𝐡 ∈ β„• ∧ 𝐴 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐡 Β· 𝐴)) = 0)
2524ancom1s 650 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐡 Β· 𝐴)) = 0)
2623, 25eqtrd 2764 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
27263adantl3 1165 . . 3 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = 0)
2815, 17, 273eqtr4rd 2775 . 2 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ (ΞΌβ€˜π΅) = 0) β†’ (ΞΌβ€˜(𝐴 Β· 𝐡)) = ((ΞΌβ€˜π΄) Β· (ΞΌβ€˜π΅)))
29 simpl1 1188 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐴 ∈ β„•)
30 simpl2 1189 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ 𝐡 ∈ β„•)
3129, 30nnmulcld 12263 . . . 4 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (𝐴 Β· 𝐡) ∈ β„•)
32 mumullem2 27031 . . . 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 26985 . . . 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 12324 . . . . . 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 13935 . . . . . . 7 (1...𝐡) ∈ Fin
38 prmssnn 16612 . . . . . . . . 9 β„™ βŠ† β„•
39 rabss2 4068 . . . . . . . . 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 16260 . . . . . . . . 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 3986 . . . . . . 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 9170 . . . . . . 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 14314 . . . . . 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 13935 . . . . . . 7 (1...𝐴) ∈ Fin
49 rabss2 4068 . . . . . . . . 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 16260 . . . . . . . . 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 3986 . . . . . . 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 9170 . . . . . . 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 14314 . . . . . 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 14111 . . . 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 1188 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„•)
6160nnzd 12583 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„€)
6261adantlr 712 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝐴 ∈ β„€)
63 simpl2 1189 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„•)
6463nnzd 12583 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„€)
6564adantlr 712 . . . . . . . . . 10 ((((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) ∧ 𝑝 ∈ β„™) β†’ 𝐡 ∈ β„€)
66 euclemma 16649 . . . . . . . . . 10 ((𝑝 ∈ β„™ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝑝 βˆ₯ (𝐴 Β· 𝐡) ↔ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)))
6759, 62, 65, 66syl3anc 1368 . . . . . . . . 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 3431 . . . . . . . 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 4298 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} βˆͺ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∨ 𝑝 βˆ₯ 𝐡)}
7068, 69eqtr4di 2782 . . . . . . 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 6886 . . . . . 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 4299 . . . . . . . 8 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴} ∩ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐡}) = {𝑝 ∈ β„™ ∣ (𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡)}
73 nprmdvds1 16642 . . . . . . . . . . . 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 16611 . . . . . . . . . . . . . 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 16485 . . . . . . . . . . . . 13 ((𝑝 ∈ β„€ ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ ((𝑝 βˆ₯ 𝐴 ∧ 𝑝 βˆ₯ 𝐡) β†’ 𝑝 βˆ₯ (𝐴 gcd 𝐡)))
7876, 62, 65, 77syl3anc 1368 . . . . . . . . . . . 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 1211 . . . . . . . . . . . . 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 5151 . . . . . . . . . . . 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 3138 . . . . . . . . 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 4377 . . . . . . . . 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 2776 . . . . . . 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 14340 . . . . . . 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 1368 . . . . . 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 2764 . . . . 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 7418 . . . 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 768 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΄) β‰  0)
92 muval2 26985 . . . . . 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 770 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ (𝐴 gcd 𝐡) = 1) ∧ ((ΞΌβ€˜π΄) β‰  0 ∧ (ΞΌβ€˜π΅) β‰  0)) β†’ (ΞΌβ€˜π΅) β‰  0)
95 muval2 26985 . . . . . 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 7420 . . . 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 2775 . . 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 2767 . 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 3022 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 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053  {crab 3424   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Fincfn 8936  β„‚cc 11105  0cc0 11107  1c1 11108   + caddc 11110   Β· cmul 11112  -cneg 11443  β„•cn 12210  β„•0cn0 12470  β„€cz 12556  ...cfz 13482  β†‘cexp 14025  β™―chash 14288   βˆ₯ cdvds 16196   gcd cgcd 16434  β„™cprime 16607  ΞΌcmu 26946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-dju 9893  df-card 9931  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-q 12931  df-rp 12973  df-fz 13483  df-fl 13755  df-mod 13833  df-seq 13965  df-exp 14026  df-hash 14289  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-dvds 16197  df-gcd 16435  df-prm 16608  df-pc 16771  df-mu 26952
This theorem is referenced by: (None)
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