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Theorem isercolllem2 15550

Description: Lemma for isercoll 15552. (Contributed by Mario Carneiro, 6-Apr-2015.)

**Hypotheses**
Ref	Expression
isercoll.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
isercoll.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
isercoll.g	⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
isercoll.i	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))

**Assertion**
Ref	Expression
isercolllem2	⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))) = (^◡𝐺 “ (𝑀...𝑁)))

Distinct variable groups: 𝑘,𝑁 𝜑,𝑘 𝑘,𝐺 𝑘,𝑀 Allowed substitution hint: 𝑍(𝑘)

Proof of Theorem isercolllem2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn 13470	. . . . . . . 8 ⊢ (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈ ℕ)
2	1	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈ ℕ))
3		cnvimass 6033	. . . . . . . . 9 ⊢ (^◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
4		isercoll.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
6	3, 5	fssdm 6688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
7	6	sseld 3943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ))
8		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
9		nnuz 12806	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2848	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ_≥‘1))
11		ltso 11235	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → < Or ℝ)
13		fzfid 13878	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin)
14		ffun 6671	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
15		funimacnv 6582	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
16	5, 14, 15	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
17		inss1 4188	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
18	16, 17	eqsstrdi 3998	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
19	13, 18	ssfid 9211	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin)
20		ssid 3966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ ⊆ ℕ
21		isercoll.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑍 = (ℤ_≥‘𝑀)
22		isercoll.m	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23		isercoll.i	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
24	21, 22, 4, 23	isercolllem1 15549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25	20, 24	mpan2 689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26		ffn 6668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
27		fnresdm 6620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28		isoeq1 7262	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
29	4, 26, 27, 28	4syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
30	25, 29	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
31		isof1o 7268	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32		f1ocnv 6796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33		f1ofun 6786	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ^◡𝐺)
34	30, 31, 32, 33	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ^◡𝐺)
35		df-f1 6501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ^◡𝐺))
36	4, 34, 35	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍)
37	36	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ–1-1→𝑍)
38		nnex 12159	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
39		ssexg 5280	. . . . . . . . . . . . . . . . 17 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
40	6, 38, 39	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
41		f1imaeng 8954	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ℕ–1-1→𝑍 ∧ (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ (^◡𝐺 “ (𝑀...𝑁)) ∈ V) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ≈ (^◡𝐺 “ (𝑀...𝑁)))
42	37, 6, 40, 41	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ≈ (^◡𝐺 “ (𝑀...𝑁)))
43	42	ensymd 8945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))
44		enfii 9133	. . . . . . . . . . . . . 14 ⊢ (((𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
45	19, 43, 44	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
46		1nn 12164	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ ℕ)
48		ffvelcdm 7032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
49	4, 46, 48	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
50	49, 21	eleqtrdi 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
51	50	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
52		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ_≥‘(𝐺‘1)))
53		elfzuzb 13435	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))))
54	51, 52, 53	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
55	5	ffnd 6669	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺 Fn ℕ)
56		elpreima 7008	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 Fn ℕ → (1 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
57	55, 56	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
58	47, 54, 57	mpbir2and 711	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ (^◡𝐺 “ (𝑀...𝑁)))
59	58	ne0d 4295	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
60		nnssre 12157	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℝ
61	6, 60	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)
62		fisupcl 9405	. . . . . . . . . . . . 13 ⊢ (( < Or ℝ ∧ ((^◡𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)))
63	12, 45, 59, 61, 62	syl13anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)))
64	6, 63	sseldd 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
65	64	nnzd 12526	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ)
66		elfz5 13433	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ_≥‘1) ∧ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
67	10, 65, 66	syl2anr 597	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
68		elpreima 7008	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn ℕ → (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁))))
69	55, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁))))
70	63, 69	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))
71		elfzle2 13445	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
72	70, 71	simpl2im 504	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
73	72	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
74		uzssz 12784	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
75	21, 74	eqsstri 3978	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 ⊆ ℤ
76		zssre 12506	. . . . . . . . . . . . . . . 16 ⊢ ℤ ⊆ ℝ
77	75, 76	sstri 3953	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℝ
78	5	ffvelcdmda 7035	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍)
79	77, 78	sselid 3942	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
80	5, 64	ffvelcdmd 7036	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
81	80	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
82	77, 81	sselid 3942	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ)
83		eluzelz 12773	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐺‘1)) → 𝑁 ∈ ℤ)
84	83	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
85	76, 84	sselid 3942	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
86		letr 11249	. . . . . . . . . . . . . 14 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺‘𝑥) ≤ 𝑁))
87	79, 82, 85, 86	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺‘𝑥) ≤ 𝑁))
88	73, 87	mpan2d 692	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺‘𝑥) ≤ 𝑁))
89	30	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
90	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ)
91		ressxr 11199	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
92	90, 91	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ^*)
93		imassrn 6024	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 “ ℕ) ⊆ ran 𝐺
94	4	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
95	94	frnd 6676	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ 𝑍)
96	93, 95	sstrid 3955	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍)
97	96, 77	sstrdi 3956	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
98	97, 91	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ^*)
99		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
100	64	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
101		leisorel 14359	. . . . . . . . . . . . 13 ⊢ ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (ℕ ⊆ ℝ^* ∧ (𝐺 “ ℕ) ⊆ ℝ^*) ∧ (𝑥 ∈ ℕ ∧ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ (𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))))
102	89, 92, 98, 99, 100, 101	syl122anc 1379	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ (𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))))
103	78, 21	eleqtrdi 2848	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ_≥‘𝑀))
104		elfz5 13433	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑥) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
105	103, 84, 104	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
106	88, 102, 105	3imtr4d 293	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → (𝐺‘𝑥) ∈ (𝑀...𝑁)))
107		elpreima 7008	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn ℕ → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...𝑁))))
108	107	baibd 540	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn ℕ ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
109	55, 108	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
110	106, 109	sylibrd 258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
111		fimaxre2 12100	. . . . . . . . . . . . 13 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
112	61, 45, 111	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
113		suprub 12116	. . . . . . . . . . . . 13 ⊢ ((((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) ∧ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))
114	113	ex 413	. . . . . . . . . . . 12 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
115	61, 59, 112, 114	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
116	115	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
117	110, 116	impbid 211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
118	67, 117	bitrd 278	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
119	118	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ ℕ → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)))))
120	2, 7, 119	pm5.21ndd 380	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
121	120	eqrdv 2734	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (^◡𝐺 “ (𝑀...𝑁)))
122	121	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (♯‘(^◡𝐺 “ (𝑀...𝑁))))
123	64	nnnn0d 12473	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ₀)
124		hashfz1 14246	. . . . 5 ⊢ (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ₀ → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))
125	123, 124	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))
126		hashen 14247	. . . . . 6 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin) → ((♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) ↔ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
127	45, 19, 126	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ((♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) ↔ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
128	43, 127	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
129	122, 125, 128	3eqtr3d 2784	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
130	129	oveq2d 7373	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))))
131	130, 121	eqtr3d 2778	1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))) = (^◡𝐺 “ (𝑀...𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 Or wor 5544 ^◡ccnv 5632 ran crn 5634 ↾ cres 5635 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 –1-1→wf1 6493 –1-1-onto→wf1o 6495 ‘cfv 6496 Isom wiso 6497 (class class class)co 7357 ≈ cen 8880 Fincfn 8883 supcsup 9376 ℝcr 11050 1c1 11052 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: isercolllem3 15551

W3C validator