Theorem lgsfval 25880

Description: Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypothesis

Ref	Expression
lgsval.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))

Assertion

Ref	Expression
lgsfval	⊢ (𝑀 ∈ ℕ → (𝐹‘𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))

Distinct variable groups:   𝐴,𝑛   𝑛,𝑀   𝑛,𝑁

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem lgsfval

Step	Hyp	Ref	Expression
1		eleq1 2902	. . 3 ⊢ (𝑛 = 𝑀 → (𝑛 ∈ ℙ ↔ 𝑀 ∈ ℙ))
2		eqeq1 2827	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑛 = 2 ↔ 𝑀 = 2))
3		oveq1 7165	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝑛 − 1) = (𝑀 − 1))
4	3	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝑛 − 1) / 2) = ((𝑀 − 1) / 2))
5	4	oveq2d 7174	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝐴↑((𝑛 − 1) / 2)) = (𝐴↑((𝑀 − 1) / 2)))
6	5	oveq1d 7173	. . . . . . 7 ⊢ (𝑛 = 𝑀 → ((𝐴↑((𝑛 − 1) / 2)) + 1) = ((𝐴↑((𝑀 − 1) / 2)) + 1))
7		id 22	. . . . . . 7 ⊢ (𝑛 = 𝑀 → 𝑛 = 𝑀)
8	6, 7	oveq12d 7176	. . . . . 6 ⊢ (𝑛 = 𝑀 → (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀))
9	8	oveq1d 7173	. . . . 5 ⊢ (𝑛 = 𝑀 → ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) = ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))
10	2, 9	ifbieq2d 4494	. . . 4 ⊢ (𝑛 = 𝑀 → if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1)) = if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1)))
11		oveq1 7165	. . . 4 ⊢ (𝑛 = 𝑀 → (𝑛 pCnt 𝑁) = (𝑀 pCnt 𝑁))
12	10, 11	oveq12d 7176	. . 3 ⊢ (𝑛 = 𝑀 → (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)) = (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)))
13	1, 12	ifbieq1d 4492	. 2 ⊢ (𝑛 = 𝑀 → if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))
14		lgsval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
15		ovex 7191	. . 3 ⊢ (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)) ∈ V
16		1ex 10639	. . 3 ⊢ 1 ∈ V
17	15, 16	ifex 4517	. 2 ⊢ if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1) ∈ V
18	13, 14, 17	fvmpt 6770	1 ⊢ (𝑀 ∈ ℕ → (𝐹‘𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ifcif 4469  {cpr 4571   class class class wbr 5068   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158  0cc0 10539  1c1 10540   + caddc 10542   − cmin 10872  -cneg 10873   / cdiv 11299  ℕcn 11640  2c2 11695  7c7 11700  8c8 11701   mod cmo 13240  ↑cexp 13432   ∥ cdvds 15609  ℙcprime 16017   pCnt cpc 16175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-1cn 10597

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161

This theorem is referenced by:  lgsval2lem  25885