hashscontpow - Mathbox for metakunt

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Theorem hashscontpow 42740

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 12595	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		hashscontpow.5	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
5		odzcl 16830	. . . . 5 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
6	1, 3, 4, 5	syl3anc 1391	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
7	6	nnnn0d 12543	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
8		hashfz1 14360	. . 3 ⊢ (((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀ → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) = ((od_ℤ‘𝑅)‘𝑁))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) = ((od_ℤ‘𝑅)‘𝑁))
10		ovexd 7432	. . . 4 ⊢ (𝜑 → (1...((od_ℤ‘𝑅)‘𝑁)) ∈ V)
11	10	mptexd 7209	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V)
12		hashscontpow.6	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	12	fvexi 6882	. . . . 5 ⊢ 𝐿 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
15		imaexg 7895	. . . 4 ⊢ (𝐿 ∈ V → (𝐿 “ 𝐸) ∈ V)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐿 “ 𝐸) ∈ V)
17	1	nnnn0d 12543	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
18		hashscontpow.7	. . . . . . . . . . . 12 ⊢ 𝑌 = (ℤ/nℤ‘𝑅)
19	18	zncrng 21597	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ₀ → 𝑌 ∈ CRing)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ CRing)
21		crngring 20296	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
22	12	zrhrhm 21564	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21506	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
24		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
25	23, 24	rhmf 20534	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
26	20, 21, 22, 25	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
27	26	ffnd 6693	. . . . . . . 8 ⊢ (𝜑 → 𝐿 Fn ℤ)
28	27	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝐿 Fn ℤ)
29	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑁 ∈ ℤ)
30		elfznn 13559	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑥 ∈ ℕ)
31	30	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ)
32	31	nnnn0d 12543	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ₀)
33	29, 32	zexpcld 14101	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ ℤ)
34		oveq2 7405	. . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
38	35, 37, 32	rspcdva 3583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ 𝐸)
39	28, 33, 38	fnfvimad 7219	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝐿‘(𝑁↑𝑥)) ∈ (𝐿 “ 𝐸))
40	39	fmpttd 7097	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸))
41	2	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑁 ∈ ℕ)
42		simpllr 785	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
43		simplr 778	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑅 ∈ ℕ)
45	4	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝑁 gcd 𝑅) = 1)
46		simpr 488	. . . . . . . . . . . . . . . 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 42739	. . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑁 ∈ ℕ)
49		simplr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
50		simpllr 785	. . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑅 ∈ ℕ)
52	4	ad3antrrr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝑁 gcd 𝑅) = 1)
53		simpr 488	. . . . . . . . . . . . . . . . 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 42739	. . . . . . . . . . . . . . . 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 970	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 264	. . . . . . . . . . . . . . . 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 13530	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑎 ∈ ℤ)
61	60	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
62	61	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
63	62	zred 12678	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℝ)
64		elfzelz 13530	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℤ)
65	64	zred 12678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℝ)
66	65	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑏 ∈ ℝ)
67		lttri2 11266	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
68	63, 66, 67	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
69	59, 68	bitrd 281	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
70	69	imbi1d 343	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 410	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → 𝑥 = 𝑎)
75	74	oveq2d 7413	. . . . . . . . . . . . . 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 6872	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
78		fvexd 6883	. . . . . . . . . . . . 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 6984	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → 𝑥 = 𝑏)
81	80	oveq2d 7413	. . . . . . . . . . . . . 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 6872	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
84		fvexd 6883	. . . . . . . . . . . . 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 6984	. . . . . . . . . . . 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 259	. . . . . . . . . 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 416	. . . . . . . 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 115	. . . . . . 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 3155	. . . . . 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 3155	. . . . 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 519	. . . 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 7239	. . . 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 236	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸))
96		hashf1dmcdm 14458	. . 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 1391	. 2 ⊢ (𝜑 → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
98	9, 97	eqbrtrrd 5125	1 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 Vcvv 3455 ⊆ wss 3905 class class class wbr 5101 ↦ cmpt 5182 “ cima 5651 Fn wfn 6517 ⟶wf 6518 –1-1→wf1 6519 ‘cfv 6522 (class class class)co 7397 ℝcr 11073 1c1 11075 < clt 11217 ≤ cle 11218 ℕcn 12211 ℕ₀cn0 12482 ℤcz 12569 ...cfz 13513 ↑cexp 14075 ♯chash 14344 gcd cgcd 16529 od_ℤcodz 16799 Basecbs 17246 Ringcrg 20284 CRingccrg 20285 RingHom crh 20519 ℤ_ringczring 21499 ℤRHomczrh 21552 ℤ/nℤczn 21555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-oadd 8442 df-er 8679 df-ec 8681 df-qs 8685 df-map 8811 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-sup 9389 df-inf 9390 df-dju 9860 df-card 9898 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12483 df-xnn0 12556 df-z 12570 df-dec 12690 df-uz 12841 df-rp 12995 df-fz 13514 df-fzo 13661 df-fl 13803 df-mod 13881 df-seq 14016 df-exp 14076 df-hash 14345 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-dvds 16288 df-gcd 16530 df-prm 16707 df-odz 16801 df-phi 16802 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-0g 17471 df-imas 17539 df-qus 17540 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-mhm 18818 df-grp 18979 df-minusg 18980 df-sbg 18981 df-mulg 19111 df-subg 19166 df-nsg 19167 df-eqg 19168 df-ghm 19255 df-cmn 19823 df-abl 19824 df-mgp 20188 df-rng 20200 df-ur 20233 df-ring 20286 df-cring 20287 df-oppr 20387 df-dvdsr 20407 df-rhm 20522 df-subrng 20597 df-subrg 20621 df-lmod 20930 df-lss 21000 df-lsp 21040 df-sra 21241 df-rgmod 21242 df-lidl 21279 df-rsp 21280 df-2idl 21321 df-cnfld 21426 df-zring 21500 df-zrh 21556 df-zn 21559

This theorem is referenced by: aks6d1c3 42741

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