hashscontpow - Mathbox for metakunt

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Theorem hashscontpow 42315

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 12512	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		hashscontpow.5	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
5		odzcl 16719	. . . . 5 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
6	1, 3, 4, 5	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
7	6	nnnn0d 12460	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
8		hashfz1 14267	. . 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 7391	. . . 4 ⊢ (𝜑 → (1...((od_ℤ‘𝑅)‘𝑁)) ∈ V)
11	10	mptexd 7168	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V)
12		hashscontpow.6	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	12	fvexi 6846	. . . . 5 ⊢ 𝐿 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
15		imaexg 7853	. . . 4 ⊢ (𝐿 ∈ V → (𝐿 “ 𝐸) ∈ V)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐿 “ 𝐸) ∈ V)
17	1	nnnn0d 12460	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
18		hashscontpow.7	. . . . . . . . . . . 12 ⊢ 𝑌 = (ℤ/nℤ‘𝑅)
19	18	zncrng 21497	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ₀ → 𝑌 ∈ CRing)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ CRing)
21		crngring 20178	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
22	12	zrhrhm 21464	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21406	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
24		eqid 2734	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
25	23, 24	rhmf 20418	. . . . . . . . . 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 6661	. . . . . . . 8 ⊢ (𝜑 → 𝐿 Fn ℤ)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝐿 Fn ℤ)
29	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑁 ∈ ℤ)
30		elfznn 13467	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑥 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ)
32	31	nnnn0d 12460	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ₀)
33	29, 32	zexpcld 14008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ ℤ)
34		oveq2 7364	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑁↑𝑘) = (𝑁↑𝑥))
35	34	eleq1d 2819	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → ((𝑁↑𝑘) ∈ 𝐸 ↔ (𝑁↑𝑥) ∈ 𝐸))
36		hashscontpow.3	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → ∀𝑘 ∈ ℕ₀ (𝑁↑𝑘) ∈ 𝐸)
38	35, 37, 32	rspcdva 3575	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ 𝐸)
39	28, 33, 38	fnfvimad 7178	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝐿‘(𝑁↑𝑥)) ∈ (𝐿 “ 𝐸))
40	39	fmpttd 7058	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸))
41	2	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑁 ∈ ℕ)
42		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
43		simplr 768	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑅 ∈ ℕ)
45	4	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝑁 gcd 𝑅) = 1)
46		simpr 484	. . . . . . . . . . . . . . . 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 42314	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑁 ∈ ℕ)
49		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
50		simpllr 775	. . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑅 ∈ ℕ)
52	4	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝑁 gcd 𝑅) = 1)
53		simpr 484	. . . . . . . . . . . . . . . . 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 42314	. . . . . . . . . . . . . . . 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 2985	. . . . . . . . . . . . . . 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 959	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 262	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑎 = 𝑏 ↔ 𝑎 = 𝑏))
59	58	necon3bbid 2967	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ 𝑎 ≠ 𝑏))
60		elfzelz 13438	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑎 ∈ ℤ)
61	60	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
62	61	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
63	62	zred 12594	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℝ)
64		elfzelz 13438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℤ)
65	64	zred 12594	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → 𝑏 ∈ ℝ)
67		lttri2 11213	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
68	63, 66, 67	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
69	59, 68	bitrd 279	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
70	69	imbi1d 341	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
73		eqidd 2735	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → 𝑥 = 𝑎)
75	74	oveq2d 7372	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
78		fvexd 6847	. . . . . . . . . . . . 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 6946	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → 𝑥 = 𝑏)
81	80	oveq2d 7372	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ (1...((od_ℤ‘𝑅)‘𝑁)))
84		fvexd 6847	. . . . . . . . . . . . 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 6946	. . . . . . . . . . . 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 2991	. . . . . . . . . . 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 257	. . . . . . . . . 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 2935	. . . . . . . . 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 412	. . . . . . . 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 3126	. . . . . 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 3126	. . . . 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 511	. . . 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 7198	. . . 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 234	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((od_ℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((od_ℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸))
96		hashf1dmcdm 14365	. . 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 1373	. 2 ⊢ (𝜑 → (♯‘(1...((od_ℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
98	9, 97	eqbrtrrd 5120	1 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 Vcvv 3438 ⊆ wss 3899 class class class wbr 5096 ↦ cmpt 5177 “ cima 5625 Fn wfn 6485 ⟶wf 6486 –1-1→wf1 6487 ‘cfv 6490 (class class class)co 7356 ℝcr 11023 1c1 11025 < clt 11164 ≤ cle 11165 ℕcn 12143 ℕ₀cn0 12399 ℤcz 12486 ...cfz 13421 ↑cexp 13982 ♯chash 14251 gcd cgcd 16419 od_ℤcodz 16688 Basecbs 17134 Ringcrg 20166 CRingccrg 20167 RingHom crh 20403 ℤ_ringczring 21399 ℤRHomczrh 21452 ℤ/nℤczn 21455

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-ec 8635 df-qs 8639 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-gcd 16420 df-prm 16597 df-odz 16690 df-phi 16691 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-0g 17359 df-imas 17427 df-qus 17428 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-nsg 19052 df-eqg 19053 df-ghm 19140 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20271 df-dvdsr 20291 df-rhm 20406 df-subrng 20477 df-subrg 20501 df-lmod 20811 df-lss 20881 df-lsp 20921 df-sra 21123 df-rgmod 21124 df-lidl 21161 df-rsp 21162 df-2idl 21203 df-cnfld 21308 df-zring 21400 df-zrh 21456 df-zn 21459

This theorem is referenced by: aks6d1c3 42316

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