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Theorem smumullem 16429
Description: Lemma for smumul 16430. (Contributed by Mario Carneiro, 22-Sep-2016.)
Hypotheses
Ref Expression
smumullem.a (πœ‘ β†’ 𝐴 ∈ β„€)
smumullem.b (πœ‘ β†’ 𝐡 ∈ β„€)
smumullem.n (πœ‘ β†’ 𝑁 ∈ β„•0)
Assertion
Ref Expression
smumullem (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡)))

Proof of Theorem smumullem
Dummy variables π‘˜ 𝑛 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smumullem.n . 2 (πœ‘ β†’ 𝑁 ∈ β„•0)
2 oveq2 7409 . . . . . . . . . 10 (π‘₯ = 0 β†’ (0..^π‘₯) = (0..^0))
3 fzo0 13652 . . . . . . . . . 10 (0..^0) = βˆ…
42, 3eqtrdi 2780 . . . . . . . . 9 (π‘₯ = 0 β†’ (0..^π‘₯) = βˆ…)
54ineq2d 4204 . . . . . . . 8 (π‘₯ = 0 β†’ ((bitsβ€˜π΄) ∩ (0..^π‘₯)) = ((bitsβ€˜π΄) ∩ βˆ…))
6 in0 4383 . . . . . . . 8 ((bitsβ€˜π΄) ∩ βˆ…) = βˆ…
75, 6eqtrdi 2780 . . . . . . 7 (π‘₯ = 0 β†’ ((bitsβ€˜π΄) ∩ (0..^π‘₯)) = βˆ…)
87oveq1d 7416 . . . . . 6 (π‘₯ = 0 β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (βˆ… smul (bitsβ€˜π΅)))
9 bitsss 16363 . . . . . . 7 (bitsβ€˜π΅) βŠ† β„•0
10 smu02 16424 . . . . . . 7 ((bitsβ€˜π΅) βŠ† β„•0 β†’ (βˆ… smul (bitsβ€˜π΅)) = βˆ…)
119, 10ax-mp 5 . . . . . 6 (βˆ… smul (bitsβ€˜π΅)) = βˆ…
128, 11eqtrdi 2780 . . . . 5 (π‘₯ = 0 β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = βˆ…)
13 oveq2 7409 . . . . . . . 8 (π‘₯ = 0 β†’ (2↑π‘₯) = (2↑0))
14 2cn 12283 . . . . . . . . 9 2 ∈ β„‚
15 exp0 14027 . . . . . . . . 9 (2 ∈ β„‚ β†’ (2↑0) = 1)
1614, 15ax-mp 5 . . . . . . . 8 (2↑0) = 1
1713, 16eqtrdi 2780 . . . . . . 7 (π‘₯ = 0 β†’ (2↑π‘₯) = 1)
1817oveq2d 7417 . . . . . 6 (π‘₯ = 0 β†’ (𝐴 mod (2↑π‘₯)) = (𝐴 mod 1))
1918fvoveq1d 7423 . . . . 5 (π‘₯ = 0 β†’ (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) = (bitsβ€˜((𝐴 mod 1) Β· 𝐡)))
2012, 19eqeq12d 2740 . . . 4 (π‘₯ = 0 β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) ↔ βˆ… = (bitsβ€˜((𝐴 mod 1) Β· 𝐡))))
2120imbi2d 340 . . 3 (π‘₯ = 0 β†’ ((πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡))) ↔ (πœ‘ β†’ βˆ… = (bitsβ€˜((𝐴 mod 1) Β· 𝐡)))))
22 oveq2 7409 . . . . . . 7 (π‘₯ = π‘˜ β†’ (0..^π‘₯) = (0..^π‘˜))
2322ineq2d 4204 . . . . . 6 (π‘₯ = π‘˜ β†’ ((bitsβ€˜π΄) ∩ (0..^π‘₯)) = ((bitsβ€˜π΄) ∩ (0..^π‘˜)))
2423oveq1d 7416 . . . . 5 (π‘₯ = π‘˜ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)))
25 oveq2 7409 . . . . . . 7 (π‘₯ = π‘˜ β†’ (2↑π‘₯) = (2β†‘π‘˜))
2625oveq2d 7417 . . . . . 6 (π‘₯ = π‘˜ β†’ (𝐴 mod (2↑π‘₯)) = (𝐴 mod (2β†‘π‘˜)))
2726fvoveq1d 7423 . . . . 5 (π‘₯ = π‘˜ β†’ (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)))
2824, 27eqeq12d 2740 . . . 4 (π‘₯ = π‘˜ β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) ↔ (((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡))))
2928imbi2d 340 . . 3 (π‘₯ = π‘˜ β†’ ((πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡))) ↔ (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)))))
30 oveq2 7409 . . . . . . 7 (π‘₯ = (π‘˜ + 1) β†’ (0..^π‘₯) = (0..^(π‘˜ + 1)))
3130ineq2d 4204 . . . . . 6 (π‘₯ = (π‘˜ + 1) β†’ ((bitsβ€˜π΄) ∩ (0..^π‘₯)) = ((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))))
3231oveq1d 7416 . . . . 5 (π‘₯ = (π‘˜ + 1) β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)))
33 oveq2 7409 . . . . . . 7 (π‘₯ = (π‘˜ + 1) β†’ (2↑π‘₯) = (2↑(π‘˜ + 1)))
3433oveq2d 7417 . . . . . 6 (π‘₯ = (π‘˜ + 1) β†’ (𝐴 mod (2↑π‘₯)) = (𝐴 mod (2↑(π‘˜ + 1))))
3534fvoveq1d 7423 . . . . 5 (π‘₯ = (π‘˜ + 1) β†’ (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)))
3632, 35eqeq12d 2740 . . . 4 (π‘₯ = (π‘˜ + 1) β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) ↔ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡))))
3736imbi2d 340 . . 3 (π‘₯ = (π‘˜ + 1) β†’ ((πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡))) ↔ (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)))))
38 oveq2 7409 . . . . . . 7 (π‘₯ = 𝑁 β†’ (0..^π‘₯) = (0..^𝑁))
3938ineq2d 4204 . . . . . 6 (π‘₯ = 𝑁 β†’ ((bitsβ€˜π΄) ∩ (0..^π‘₯)) = ((bitsβ€˜π΄) ∩ (0..^𝑁)))
4039oveq1d 7416 . . . . 5 (π‘₯ = 𝑁 β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)))
41 oveq2 7409 . . . . . . 7 (π‘₯ = 𝑁 β†’ (2↑π‘₯) = (2↑𝑁))
4241oveq2d 7417 . . . . . 6 (π‘₯ = 𝑁 β†’ (𝐴 mod (2↑π‘₯)) = (𝐴 mod (2↑𝑁)))
4342fvoveq1d 7423 . . . . 5 (π‘₯ = 𝑁 β†’ (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡)))
4440, 43eqeq12d 2740 . . . 4 (π‘₯ = 𝑁 β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡)) ↔ (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡))))
4544imbi2d 340 . . 3 (π‘₯ = 𝑁 β†’ ((πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘₯)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑π‘₯)) Β· 𝐡))) ↔ (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡)))))
46 smumullem.a . . . . . . . 8 (πœ‘ β†’ 𝐴 ∈ β„€)
47 zmod10 13848 . . . . . . . 8 (𝐴 ∈ β„€ β†’ (𝐴 mod 1) = 0)
4846, 47syl 17 . . . . . . 7 (πœ‘ β†’ (𝐴 mod 1) = 0)
4948oveq1d 7416 . . . . . 6 (πœ‘ β†’ ((𝐴 mod 1) Β· 𝐡) = (0 Β· 𝐡))
50 smumullem.b . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ β„€)
5150zcnd 12663 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ β„‚)
5251mul02d 11408 . . . . . 6 (πœ‘ β†’ (0 Β· 𝐡) = 0)
5349, 52eqtrd 2764 . . . . 5 (πœ‘ β†’ ((𝐴 mod 1) Β· 𝐡) = 0)
5453fveq2d 6885 . . . 4 (πœ‘ β†’ (bitsβ€˜((𝐴 mod 1) Β· 𝐡)) = (bitsβ€˜0))
55 0bits 16376 . . . 4 (bitsβ€˜0) = βˆ…
5654, 55eqtr2di 2781 . . 3 (πœ‘ β†’ βˆ… = (bitsβ€˜((𝐴 mod 1) Β· 𝐡)))
57 oveq1 7408 . . . . . 6 ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}) = ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
58 bitsss 16363 . . . . . . . . 9 (bitsβ€˜π΄) βŠ† β„•0
5958a1i 11 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (bitsβ€˜π΄) βŠ† β„•0)
609a1i 11 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (bitsβ€˜π΅) βŠ† β„•0)
61 simpr 484 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
6259, 60, 61smup1 16426 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
63 bitsinv1lem 16378 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ π‘˜ ∈ β„•0) β†’ (𝐴 mod (2↑(π‘˜ + 1))) = ((𝐴 mod (2β†‘π‘˜)) + if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))
6446, 63sylan 579 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (𝐴 mod (2↑(π‘˜ + 1))) = ((𝐴 mod (2β†‘π‘˜)) + if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))
6564oveq1d 7416 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡) = (((𝐴 mod (2β†‘π‘˜)) + if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)) Β· 𝐡))
6646adantr 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐴 ∈ β„€)
67 2nn 12281 . . . . . . . . . . . . . . 15 2 ∈ β„•
6867a1i 11 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 2 ∈ β„•)
6968, 61nnexpcld 14204 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (2β†‘π‘˜) ∈ β„•)
7066, 69zmodcld 13853 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (𝐴 mod (2β†‘π‘˜)) ∈ β„•0)
7170nn0cnd 12530 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (𝐴 mod (2β†‘π‘˜)) ∈ β„‚)
7269nnnn0d 12528 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (2β†‘π‘˜) ∈ β„•0)
73 0nn0 12483 . . . . . . . . . . . . 13 0 ∈ β„•0
74 ifcl 4565 . . . . . . . . . . . . 13 (((2β†‘π‘˜) ∈ β„•0 ∧ 0 ∈ β„•0) β†’ if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) ∈ β„•0)
7572, 73, 74sylancl 585 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) ∈ β„•0)
7675nn0cnd 12530 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) ∈ β„‚)
7751adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐡 ∈ β„‚)
7871, 76, 77adddird 11235 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (((𝐴 mod (2β†‘π‘˜)) + if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)) Β· 𝐡) = (((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) Β· 𝐡)))
7976, 77mulcomd 11231 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) Β· 𝐡) = (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))
8079oveq2d 7417 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) Β· 𝐡)) = (((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0))))
8165, 78, 803eqtrd 2768 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡) = (((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0))))
8281fveq2d 6885 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)) = (bitsβ€˜(((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))))
8370nn0zd 12580 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (𝐴 mod (2β†‘π‘˜)) ∈ β„€)
8450adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐡 ∈ β„€)
8583, 84zmulcld 12668 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) ∈ β„€)
8675nn0zd 12580 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) ∈ β„€)
8784, 86zmulcld 12668 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)) ∈ β„€)
88 sadadd 16404 . . . . . . . . 9 ((((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) ∈ β„€ ∧ (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)) ∈ β„€) β†’ ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))) = (bitsβ€˜(((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))))
8985, 87, 88syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))) = (bitsβ€˜(((𝐴 mod (2β†‘π‘˜)) Β· 𝐡) + (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))))
90 oveq2 7409 . . . . . . . . . . 11 ((2β†‘π‘˜) = if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) β†’ (𝐡 Β· (2β†‘π‘˜)) = (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))
9190fveqeq2d 6889 . . . . . . . . . 10 ((2β†‘π‘˜) = if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) β†’ ((bitsβ€˜(𝐡 Β· (2β†‘π‘˜))) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))} ↔ (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0))) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
92 oveq2 7409 . . . . . . . . . . 11 (0 = if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) β†’ (𝐡 Β· 0) = (𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))
9392fveqeq2d 6889 . . . . . . . . . 10 (0 = if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0) β†’ ((bitsβ€˜(𝐡 Β· 0)) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))} ↔ (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0))) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
94 bitsshft 16412 . . . . . . . . . . . 12 ((𝐡 ∈ β„€ ∧ π‘˜ ∈ β„•0) β†’ {𝑛 ∈ β„•0 ∣ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)} = (bitsβ€˜(𝐡 Β· (2β†‘π‘˜))))
9550, 94sylan 579 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ {𝑛 ∈ β„•0 ∣ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)} = (bitsβ€˜(𝐡 Β· (2β†‘π‘˜))))
96 ibar 528 . . . . . . . . . . . 12 (π‘˜ ∈ (bitsβ€˜π΄) β†’ ((𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅) ↔ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))))
9796rabbidv 3432 . . . . . . . . . . 11 (π‘˜ ∈ (bitsβ€˜π΄) β†’ {𝑛 ∈ β„•0 ∣ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)} = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))})
9895, 97sylan9req 2785 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ π‘˜ ∈ (bitsβ€˜π΄)) β†’ (bitsβ€˜(𝐡 Β· (2β†‘π‘˜))) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))})
9977adantr 480 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ 𝐡 ∈ β„‚)
10099mul01d 11409 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ (𝐡 Β· 0) = 0)
101100fveq2d 6885 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ (bitsβ€˜(𝐡 Β· 0)) = (bitsβ€˜0))
102 simpr 484 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ Β¬ π‘˜ ∈ (bitsβ€˜π΄))
103102intnanrd 489 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ Β¬ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)))
104103ralrimivw 3142 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ βˆ€π‘› ∈ β„•0 Β¬ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)))
105 rabeq0 4376 . . . . . . . . . . . 12 ({𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))} = βˆ… ↔ βˆ€π‘› ∈ β„•0 Β¬ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅)))
106104, 105sylibr 233 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))} = βˆ…)
10755, 101, 1063eqtr4a 2790 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•0) ∧ Β¬ π‘˜ ∈ (bitsβ€˜π΄)) β†’ (bitsβ€˜(𝐡 Β· 0)) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))})
10891, 93, 98, 107ifbothda 4558 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0))) = {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))})
109108oveq2d 7417 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd (bitsβ€˜(𝐡 Β· if(π‘˜ ∈ (bitsβ€˜π΄), (2β†‘π‘˜), 0)))) = ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
11082, 89, 1093eqtr2d 2770 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)) = ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}))
11162, 110eqeq12d 2740 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)) ↔ ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))}) = ((bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) sadd {𝑛 ∈ β„•0 ∣ (π‘˜ ∈ (bitsβ€˜π΄) ∧ (𝑛 βˆ’ π‘˜) ∈ (bitsβ€˜π΅))})))
11257, 111imbitrrid 245 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) β†’ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡))))
113112expcom 413 . . . 4 (π‘˜ ∈ β„•0 β†’ (πœ‘ β†’ ((((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡)) β†’ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)))))
114113a2d 29 . . 3 (π‘˜ ∈ β„•0 β†’ ((πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^π‘˜)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2β†‘π‘˜)) Β· 𝐡))) β†’ (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^(π‘˜ + 1))) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑(π‘˜ + 1))) Β· 𝐡)))))
11521, 29, 37, 45, 56, 114nn0ind 12653 . 2 (𝑁 ∈ β„•0 β†’ (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡))))
1161, 115mpcom 38 1 (πœ‘ β†’ (((bitsβ€˜π΄) ∩ (0..^𝑁)) smul (bitsβ€˜π΅)) = (bitsβ€˜((𝐴 mod (2↑𝑁)) Β· 𝐡)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  {crab 3424   ∩ cin 3939   βŠ† wss 3940  βˆ…c0 4314  ifcif 4520  β€˜cfv 6533  (class class class)co 7401  β„‚cc 11103  0cc0 11105  1c1 11106   + caddc 11108   Β· cmul 11110   βˆ’ cmin 11440  β„•cn 12208  2c2 12263  β„•0cn0 12468  β„€cz 12554  ..^cfzo 13623   mod cmo 13830  β†‘cexp 14023  bitscbits 16356   sadd csad 16357   smul csmu 16358
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-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-had 1587  df-cad 1600  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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-disj 5104  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-sup 9432  df-inf 9433  df-oi 9500  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-dvds 16194  df-bits 16359  df-sad 16388  df-smu 16413
This theorem is referenced by:  smumul  16430
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