Theorem hashscontpow 42099

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 → ((odℤ‘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 12498	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		hashscontpow.5	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
5		odzcl 16705	. . . . 5 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) → ((odℤ‘𝑅)‘𝑁) ∈ ℕ)
6	1, 3, 4, 5	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((odℤ‘𝑅)‘𝑁) ∈ ℕ)
7	6	nnnn0d 12445	. . 3 ⊢ (𝜑 → ((odℤ‘𝑅)‘𝑁) ∈ ℕ0)
8		hashfz1 14253	. . 3 ⊢ (((odℤ‘𝑅)‘𝑁) ∈ ℕ0 → (♯‘(1...((odℤ‘𝑅)‘𝑁))) = ((odℤ‘𝑅)‘𝑁))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (♯‘(1...((odℤ‘𝑅)‘𝑁))) = ((odℤ‘𝑅)‘𝑁))
10		ovexd 7384	. . . 4 ⊢ (𝜑 → (1...((odℤ‘𝑅)‘𝑁)) ∈ V)
11	10	mptexd 7160	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V)
12		hashscontpow.6	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	12	fvexi 6836	. . . . 5 ⊢ 𝐿 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
15		imaexg 7846	. . . 4 ⊢ (𝐿 ∈ V → (𝐿 “ 𝐸) ∈ V)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐿 “ 𝐸) ∈ V)
17	1	nnnn0d 12445	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
18		hashscontpow.7	. . . . . . . . . . . 12 ⊢ 𝑌 = (ℤ/nℤ‘𝑅)
19	18	zncrng 21451	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ0 → 𝑌 ∈ CRing)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ CRing)
21		crngring 20130	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
22	12	zrhrhm 21418	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
23		zringbas 21360	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
24		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
25	23, 24	rhmf 20370	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
26	20, 21, 22, 25	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
27	26	ffnd 6653	. . . . . . . 8 ⊢ (𝜑 → 𝐿 Fn ℤ)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝐿 Fn ℤ)
29	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑁 ∈ ℤ)
30		elfznn 13456	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) → 𝑥 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ)
32	31	nnnn0d 12445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑥 ∈ ℕ0)
33	29, 32	zexpcld 13994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ ℤ)
34		oveq2 7357	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑁↑𝑘) = (𝑁↑𝑥))
35	34	eleq1d 2813	. . . . . . . 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...((odℤ‘𝑅)‘𝑁))) → ∀𝑘 ∈ ℕ0 (𝑁↑𝑘) ∈ 𝐸)
38	35, 37, 32	rspcdva 3578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (𝑁↑𝑥) ∈ 𝐸)
39	28, 33, 38	fnfvimad 7170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (𝐿‘(𝑁↑𝑥)) ∈ (𝐿 “ 𝐸))
40	39	fmpttd 7049	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸))
41	2	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑁 ∈ ℕ)
42		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁)))
43		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁)))
44	1	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑅 ∈ ℕ)
45	4	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝑁 gcd 𝑅) = 1)
46		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
47	41, 42, 43, 44, 45, 12, 18, 46	hashscontpow1 42098	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑎 < 𝑏) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
48	2	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑁 ∈ ℕ)
49		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁)))
50		simpllr 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁)))
51	1	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑅 ∈ ℕ)
52	4	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝑁 gcd 𝑅) = 1)
53		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → 𝑏 < 𝑎)
54	48, 49, 50, 51, 52, 12, 18, 53	hashscontpow1 42098	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑏)) ≠ (𝐿‘(𝑁↑𝑎)))
55	54	necomd 2980	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
56	47, 55	jaodan 959	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
57	56	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
58		biidd 262	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (𝑎 = 𝑏 ↔ 𝑎 = 𝑏))
59	58	necon3bbid 2962	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ 𝑎 ≠ 𝑏))
60		elfzelz 13427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁)) → 𝑎 ∈ ℤ)
61	60	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
62	61	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℤ)
63	62	zred 12580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑎 ∈ ℝ)
64		elfzelz 13427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℤ)
65	64	zred 12580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁)) → 𝑏 ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → 𝑏 ∈ ℝ)
67		lttri2 11198	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
68	63, 66, 67	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
69	59, 68	bitrd 279	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
70	69	imbi1d 341	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘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...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
72	71	imp 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏)))
73		eqidd 2730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) = (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))))
74		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → 𝑥 = 𝑎)
75	74	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → (𝑁↑𝑥) = (𝑁↑𝑎))
76	75	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑎) → (𝐿‘(𝑁↑𝑥)) = (𝐿‘(𝑁↑𝑎)))
77		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁)))
78		fvexd 6837	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑎)) ∈ V)
79	73, 76, 77, 78	fvmptd 6937	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = (𝐿‘(𝑁↑𝑎)))
80		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → 𝑥 = 𝑏)
81	80	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → (𝑁↑𝑥) = (𝑁↑𝑏))
82	81	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑥 = 𝑏) → (𝐿‘(𝑁↑𝑥)) = (𝐿‘(𝑁↑𝑏)))
83		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁)))
84		fvexd 6837	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (𝐿‘(𝑁↑𝑏)) ∈ V)
85	73, 82, 83, 84	fvmptd 6937	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) = (𝐿‘(𝑁↑𝑏)))
86	79, 85	neeq12d 2986	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → (((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) ≠ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) ↔ (𝐿‘(𝑁↑𝑎)) ≠ (𝐿‘(𝑁↑𝑏))))
87	72, 86	mpbird 257	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) ≠ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏))
88	87	neneqd 2930	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ ¬ 𝑎 = 𝑏) → ¬ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏))
89	88	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (¬ 𝑎 = 𝑏 → ¬ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏)))
90	89	con4d 115	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) ∧ 𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))) → (((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
91	90	ralrimiva 3121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))) → ∀𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
92	91	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏))
93	40, 92	jca 511	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸) ∧ ∀𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏)))
94		dff13 7191	. . . 4 ⊢ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸) ↔ ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))⟶(𝐿 “ 𝐸) ∧ ∀𝑎 ∈ (1...((odℤ‘𝑅)‘𝑁))∀𝑏 ∈ (1...((odℤ‘𝑅)‘𝑁))(((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑎) = ((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥)))‘𝑏) → 𝑎 = 𝑏)))
95	93, 94	sylibr 234	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸))
96		hashf1dmcdm 14351	. . 3 ⊢ (((𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))) ∈ V ∧ (𝐿 “ 𝐸) ∈ V ∧ (𝑥 ∈ (1...((odℤ‘𝑅)‘𝑁)) ↦ (𝐿‘(𝑁↑𝑥))):(1...((odℤ‘𝑅)‘𝑁))–1-1→(𝐿 “ 𝐸)) → (♯‘(1...((odℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
97	11, 16, 95, 96	syl3anc 1373	. 2 ⊢ (𝜑 → (♯‘(1...((odℤ‘𝑅)‘𝑁))) ≤ (♯‘(𝐿 “ 𝐸)))
98	9, 97	eqbrtrrd 5116	1 ⊢ (𝜑 → ((odℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ⊆ wss 3903   class class class wbr 5092   ↦ cmpt 5173   “ cima 5622   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479  ‘cfv 6482  (class class class)co 7349  ℝcr 11008  1c1 11010   < clt 11149   ≤ cle 11150  ℕcn 12128  ℕ₀cn0 12384  ℤcz 12471  ...cfz 13410  ↑cexp 13968  ♯chash 14237   gcd cgcd 16405  od_ℤcodz 16674  Basecbs 17120  Ringcrg 20118  CRingccrg 20119   RingHom crh 20354  ℤ_ringczring 21353  ℤRHomczrh 21406  ℤ/nℤczn 21409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088  ax-mulf 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-gcd 16406  df-prm 16583  df-odz 16676  df-phi 16677  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-nsg 19003  df-eqg 19004  df-ghm 19092  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-lmod 20765  df-lss 20835  df-lsp 20875  df-sra 21077  df-rgmod 21078  df-lidl 21115  df-rsp 21116  df-2idl 21157  df-cnfld 21262  df-zring 21354  df-zrh 21410  df-zn 21413

This theorem is referenced by:  aks6d1c3  42100