hashscontpow - Mathbox for metakunt

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Theorem hashscontpow 43315

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 12732	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		hashscontpow.5	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
5		odzcl 16986	. . . . 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 12680	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
8		hashfz1 14503	. . 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 7465	. . . 4 ⊢ (𝜑 → (1...((od_ℤ‘𝑅)‘𝑁)) ∈ V)
11	10	mptexd 7240	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V)
12		hashscontpow.6	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	12	fvexi 6910	. . . . 5 ⊢ 𝐿 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
15		imaexg 7935	. . . 4 ⊢ (𝐿 ∈ V → (𝐿 “ 𝐸) ∈ V)
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → (𝐿 “ 𝐸) ∈ V)
17	1	nnnn0d 12680	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
18		hashscontpow.7	. . . . . . . . . . . 12 ⊢ 𝑌 = (ℤ/nℤ‘𝑅)
19	18	zncrng 21865	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ₀ → 𝑌 ∈ CRing)
20	17, 19	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ CRing)
21		crngring 20487	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
22	12	zrhrhm 21832	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21774	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
24		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
25	23, 24	rhmf 20730	. . . . . . . . . 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 6721	. . . . . . . 8 ⊢ (𝜑 → 𝐿 Fn ℤ)
28	27	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝐿 Fn ℤ)
29	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑁 ∈ ℤ)
30		elfznn 13700	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑥 ∈ ℕ)
31	30	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ)
32	31	nnnn0d 12680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ₀)
33	29, 32	zexpcld 14243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ ℤ)
34		oveq2 7438	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑁↑𝑘) = (𝑁↑𝑥))
35	34	eleq1d 2849	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → ((𝑁↑𝑘) ∈ 𝐸 ↔ (𝑁↑𝑥) ∈ 𝐸))
36		hashscontpow.3	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
37	36	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
38	35, 37, 32	rspcdva 3581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ 𝐸)
39	28, 33, 38	fnfvimad 7250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝐿‘(𝑁↑𝑥)) ∈ (𝐿 “ 𝐸))
40	39	fmpttd 7126	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸))
41	2	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑁 ∈ ℕ)
42		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
43		simplr 781	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑅 ∈ ℕ)
45	4	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝑁 gcd 𝑅) = 1)
46		simpr 490	. . . . . . . . . . . . . . . 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 43314	. . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑁 ∈ ℕ)
49		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
50		simpllr 788	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑅 ∈ ℕ)
52	4	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝑁 gcd 𝑅) = 1)
53		simpr 490	. . . . . . . . . . . . . . . . 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 43314	. . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 2996	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ 𝑎 ≠ 𝑏))
60		elfzelz 13669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑎 ∈ ℤ)
61	60	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
62	61	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
63	62	zred 12816	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℝ)
64		elfzelz 13669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℤ)
65	64	zred 12816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℝ)
66	65	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑏 ∈ ℝ)
67		lttri2 11405	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
68	63, 66, 67	syl2anc 596	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → 𝑥 = 𝑎)
75	74	oveq2d 7446	. . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
78		fvexd 6911	. . . . . . . . . . . . 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 7012	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → 𝑥 = 𝑏)
81	80	oveq2d 7446	. . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
84		fvexd 6911	. . . . . . . . . . . . 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 7012	. . . . . . . . . . . 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 3020	. . . . . . . . . . 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 2964	. . . . . . . . 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 418	. . . . . . . 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 3158	. . . . . 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 3158	. . . . 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 521	. . . 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 7268	. . . 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 14602	. . 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 5132	1 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 class class class wbr 5106 ↦ cmpt 5189 “ cima 5658 Fn wfn 6542 ⟶wf 6543 –1-1→wf1 6544 ‘cfv 6547 (class class class)co 7430 ℝcr 11212 1c1 11214 < clt 11356 ≤ cle 11357 ℕcn 12348 ℕ₀cn0 12619 ℤcz 12706 ...cfz 13652 ↑cexp 14217 ♯chash 14487 gcd cgcd 16679 od_ℤcodz 16955 Basecbs 17402 Ringcrg 20474 CRingccrg 20475 RingHom crh 20714 ℤ_ringczring 21767 ℤRHomczrh 21820 ℤ/nℤczn 21823

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292 ax-mulf 11293

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-oadd 8494 df-er 8731 df-ec 8733 df-qs 8737 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-sup 9447 df-inf 9448 df-dju 9995 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-xnn0 12693 df-z 12707 df-dec 12828 df-uz 12979 df-rp 13134 df-fz 13653 df-fzo 13802 df-fl 13945 df-mod 14023 df-seq 14158 df-exp 14218 df-hash 14488 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-dvds 16438 df-gcd 16680 df-prm 16862 df-odz 16957 df-phi 16958 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-0g 17627 df-imas 17695 df-qus 17696 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-mhm 18993 df-grp 19162 df-minusg 19163 df-sbg 19164 df-mulg 19293 df-subg 19348 df-nsg 19349 df-eqg 19350 df-ghm 19443 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-cring 20477 df-oppr 20582 df-dvdsr 20602 df-rhm 20717 df-subrng 20813 df-subrg 20837 df-lmod 21152 df-lss 21222 df-lsp 21262 df-sra 21463 df-rgmod 21464 df-lidl 21501 df-rsp 21502 df-2idl 21558 df-cnfld 21694 df-zring 21768 df-zrh 21824 df-zn 21827

This theorem is used by: aks6d1c3 43316

W3C validator