summolem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > summolem3		Structured version Visualization version GIF version

Theorem summolem3 15680

Description: Lemma for summo 15683. (Contributed by Mario Carneiro, 29-Mar-2014.)

**Hypotheses**
Ref	Expression
summo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
summo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summo.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
summolem3.4	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
summolem3.5	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
summolem3.6	⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
summolem3.7	⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)

**Assertion**
Ref	Expression
summolem3	⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Distinct variable groups: 𝑓,𝑘,𝑛,𝐴 𝑓,𝐹,𝑘,𝑛 𝑘,𝐺,𝑛 𝑘,𝐾,𝑛 𝑘,𝑁,𝑛 𝜑,𝑘,𝑛 𝐵,𝑓,𝑛 𝑘,𝑀,𝑛 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑘) 𝐺(𝑓) 𝐻(𝑓,𝑘,𝑛) 𝐾(𝑓) 𝑀(𝑓) 𝑁(𝑓)

Proof of Theorem summolem3 Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 11150	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 11360	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
4	3	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5		addass 11155	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
6	5	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7		summolem3.5	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 494	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 12836	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2838	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		ssidd 3970	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		summolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
13		f1ocnv 6812	. . . . . 6 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
15		summolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
16		f1oco 6823	. . . . 5 ⊢ ((^◡𝑓:𝐴–1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto→𝐴) → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17	14, 15, 16	syl2anc 584	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18		ovex 7420	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 8944	. . . . . . . . 9 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
20	17, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ≈ (1...𝑀))
21		fzfi 13937	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 13937	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14312	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 692	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27		nnnn0 12449	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
28		hashfz1 14311	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
29	26, 27, 28	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30		nnnn0 12449	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
31		hashfz1 14311	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
32	8, 30, 31	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2773	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7403	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6796	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
36	17, 35	mpbird 257	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37		summo.3	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
38		fveq2 6858	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
39	38	csbeq1d 3866	. . . . 5 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13514	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6800	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
44	43	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		summo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3125	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3886	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2908	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3876	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2813	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3576	. . . . . 6 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
53	44, 47, 52	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
54	37, 39, 41, 53	fvmptd3 6991	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2828	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6796	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
57	15, 56	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
58		f1of 6800	. . . . . . . . . 10 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
59	57, 58	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
60		fvco3 6960	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
64	59	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7252	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtr2d 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
68	67	csbeq1d 3866	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
69	68	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
70		elfznn 13514	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
71	70	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
72		fveq2 6858	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
73	72	csbeq1d 3866	. . . . . 6 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
74		summolem3.4	. . . . . 6 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
75	73, 74	fvmpti 6967	. . . . 5 ⊢ (𝑖 ∈ ℕ → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
76	71, 75	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
77		f1of 6800	. . . . . . 7 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
78	36, 77	syl 17	. . . . . 6 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
79	78	ffvelcdmda 7056	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
80		elfznn 13514	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
81		fveq2 6858	. . . . . . 7 ⊢ (𝑛 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
82	81	csbeq1d 3866	. . . . . 6 ⊢ (𝑛 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
83	82, 37	fvmpti 6967	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
84	79, 80, 83	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
85	69, 76, 84	3eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 14008	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
87	33	fveq2d 6862	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2766	1 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⦋csb 3862 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 I cid 5532 ^◡ccnv 5637 ∘ ccom 5642 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 ≈ cen 8915 Fincfn 8918 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468 seqcseq 13966 ♯chash 14295

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296

This theorem is referenced by: summolem2a 15681 summo 15683

W3C validator