hashscontpow - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hashscontpow		Structured version Visualization version GIF version

Theorem hashscontpow 42909

Description: If a set contains all 𝑁-th powers, then the size of the image under the ZR homomorphism is greater than the 𝑅-th order of 𝑁. (Contributed by metakunt, 28-Apr-2025.)

**Hypotheses**
Ref	Expression
hashscontpow.1	⊢ (𝜑 → 𝐸 ⊆ ℤ)
hashscontpow.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
hashscontpow.3	⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
hashscontpow.4	⊢ (𝜑 → 𝑅 ∈ ℕ)
hashscontpow.5	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
hashscontpow.6	⊢ 𝐿 = (ℤRHom‘𝑌)
hashscontpow.7	⊢ 𝑌 = (ℤ/nℤ‘𝑅)

**Assertion**
Ref	Expression
hashscontpow	⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Distinct variable groups: 𝑘,𝐸 𝑘,𝑁 Allowed substitution hints: 𝜑(𝑘) 𝑅(𝑘) 𝐿(𝑘) 𝑌(𝑘)

Proof of Theorem hashscontpow Dummy variables 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashscontpow.4	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
2		hashscontpow.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nnzd 12612	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		hashscontpow.5	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
5		odzcl 16848	. . . . 5 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
6	1, 3, 4, 5	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
7	6	nnnn0d 12560	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
8		hashfz1 14378	. . 3 ⊢ (((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀ → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) = ((od_ℤ‘𝑅)‘𝑁))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) = ((od_ℤ‘𝑅)‘𝑁))
10		ovexd 7445	. . . 4 ⊢ (𝜑 → (1...((od_ℤ‘𝑅)‘𝑁)) ∈ V)
11	10	mptexd 7222	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V)
12		hashscontpow.6	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	12	fvexi 6895	. . . . 5 ⊢ 𝐿 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
15		imaexg 7906	. . . 4 ⊢ (𝐿 ∈ V → (𝐿 “ 𝐸) ∈ V)
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → (𝐿 “ 𝐸) ∈ V)
17	1	nnnn0d 12560	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
18		hashscontpow.7	. . . . . . . . . . . 12 ⊢ 𝑌 = (ℤ/nℤ‘𝑅)
19	18	zncrng 21694	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ₀ → 𝑌 ∈ CRing)
20	17, 19	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ CRing)
21		crngring 20322	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
22	12	zrhrhm 21661	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21603	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
24		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
25	23, 24	rhmf 20563	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
26	20, 21, 22, 25	4syl 20	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
27	26	ffnd 6706	. . . . . . . 8 ⊢ (𝜑 → 𝐿 Fn ℤ)
28	27	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝐿 Fn ℤ)
29	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑁 ∈ ℤ)
30		elfznn 13577	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑥 ∈ ℕ)
31	30	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ)
32	31	nnnn0d 12560	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ₀)
33	29, 32	zexpcld 14119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ ℤ)
34		oveq2 7418	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑁↑𝑘) = (𝑁↑𝑥))
35	34	eleq1d 2848	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → ((𝑁↑𝑘) ∈ 𝐸 ↔ (𝑁↑𝑥) ∈ 𝐸))
36		hashscontpow.3	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
37	36	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
38	35, 37, 32	rspcdva 3582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ 𝐸)
39	28, 33, 38	fnfvimad 7232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝐿‘(𝑁↑𝑥)) ∈ (𝐿 “ 𝐸))
40	39	fmpttd 7110	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸))
41	2	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑁 ∈ ℕ)
42		simpllr 787	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
43		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
44	1	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑅 ∈ ℕ)
45	4	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝑁 gcd 𝑅) = 1)
46		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
47	41, 42, 43, 44, 45, 12, 18, 46	hashscontpow1 42908	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
48	2	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑁 ∈ ℕ)
49		simplr 780	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
50		simpllr 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
51	1	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑅 ∈ ℕ)
52	4	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝑁 gcd 𝑅) = 1)
53		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 < 𝑎)
54	48, 49, 50, 51, 52, 12, 18, 53	hashscontpow1 42908	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑏)) ≠ (𝐿‘(𝑁↑𝑎)))
55	54	necomd 3013	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
56	47, 55	jaodan 972	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
57	56	ex 417	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
58		biidd 265	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑎 = 𝑏 ↔ 𝑎 = 𝑏))
59	58	necon3bbid 2995	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ 𝑎 ≠ 𝑏))
60		elfzelz 13547	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑎 ∈ ℤ)
61	60	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
62	61	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
63	62	zred 12695	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℝ)
64		elfzelz 13547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℤ)
65	64	zred 12695	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℝ)
66	65	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑏 ∈ ℝ)
67		lttri2 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
68	63, 66, 67	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
69	59, 68	bitrd 282	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
70	69	imbi1d 344	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ((¬ 𝑎 = 𝑏 → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))) ↔ ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))))
71	57, 70	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
72	71	imp 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
73		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) = (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))))
74		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → 𝑥 = 𝑎)
75	74	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → (𝑁↑𝑥) = (𝑁↑𝑎))
76	75	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → (𝐿‘(𝑁↑𝑥)) = (𝐿‘(𝑁↑𝑎)))
77		simpllr 787	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
78		fvexd 6896	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑎)) ∈ V)
79	73, 76, 77, 78	fvmptd 6997	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = (𝐿‘(𝑁↑𝑎)))
80		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → 𝑥 = 𝑏)
81	80	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → (𝑁↑𝑥) = (𝑁↑𝑏))
82	81	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → (𝐿‘(𝑁↑𝑥)) = (𝐿‘(𝑁↑𝑏)))
83		simplr 780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
84		fvexd 6896	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑏)) ∈ V)
85	73, 82, 83, 84	fvmptd 6997	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) = (𝐿‘(𝑁↑𝑏)))
86	79, 85	neeq12d 3019	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) ≠ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) ↔ (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
87	72, 86	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) ≠ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏))
88	87	neneqd 2963	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ¬ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏))
89	88	ex 417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 → ¬ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏)))
90	89	con4d 116	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
91	90	ralrimiva 3157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ∀𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
92	91	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
93	40, 92	jca 520	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸) ∧ ∀𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏)))
94		dff13 7252	. . . 4 ⊢ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸) ↔ ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸) ∧ ∀𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏)))
95	93, 94	sylibr 237	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸))
96		hashf1dmcdm 14477	. . 3 ⊢ (((𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V ∧ (𝐿 “ 𝐸) ∈ V ∧ (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸)) → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
97	11, 16, 95, 96	syl3anc 1398	. 2 ⊢ (𝜑 → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
98	9, 97	eqbrtrrd 5135	1 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 Vcvv 3455 ⊆ wss 3905 class class class wbr 5109 ↦ cmpt 5192 “ cima 5664 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7410 ℝcr 11094 1c1 11096 < clt 11238 ≤ cle 11239 ℕcn 12228 ℕ₀cn0 12499 ℤcz 12586 ...cfz 13530 ↑cexp 14093 ♯chash 14362 gcd cgcd 16547 od_ℤcodz 16817 Basecbs 17264 Ringcrg 20310 CRingccrg 20311 RingHom crh 20547 ℤ_ringczring 21596 ℤRHomczrh 21649 ℤ/nℤczn 21652

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-prm 16725 df-odz 16819 df-phi 16820 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-0g 17489 df-imas 17557 df-qus 17558 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-nsg 19185 df-eqg 19186 df-ghm 19279 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-lmod 20983 df-lss 21053 df-lsp 21093 df-sra 21294 df-rgmod 21295 df-lidl 21332 df-rsp 21333 df-2idl 21389 df-cnfld 21523 df-zring 21597 df-zrh 21653 df-zn 21656

This theorem is referenced by: aks6d1c3 42910

W3C validator