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Theorem isercolllem2 15676

Description: Lemma for isercoll 15678. (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 13555	. . . . . . . 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 6068	. . . . . . . . 9 ⊢ (^◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
4		isercoll.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
5	4	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
6	3, 5	fssdm 6707	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
7	6	sseld 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ))
8		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
9		nnuz 12875	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2871	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ_≥‘1))
11		ltso 11260	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → < Or ℝ)
13		fzfid 13983	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin)
14		ffun 6690	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
15		funimacnv 6598	. . . . . . . . . . . . . . . . 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 3980	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
19	13, 18	ssfid 9209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin)
20		ssid 3958	. . . . . . . . . . . . . . . . . . . . 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 15675	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25	20, 24	mpan2 701	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26		ffn 6687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
27		fnresdm 6636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28		isoeq1 7297	. . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
31		isof1o 7303	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32		f1ocnv 6815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33		f1ofun 6804	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ^◡𝐺)
34	30, 31, 32, 33	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ^◡𝐺)
35		df-f1 6522	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ^◡𝐺))
36	4, 34, 35	sylanbrc 592	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍)
37	36	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ–1-1→𝑍)
38		nnex 12213	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
39		ssexg 5278	. . . . . . . . . . . . . . . . 17 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
40	6, 38, 39	sylancl 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
41		f1imaeng 8991	. . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ≈ (^◡𝐺 “ (𝑀...𝑁)))
43	42	ensymd 8982	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))
44		enfii 9150	. . . . . . . . . . . . . 14 ⊢ (((𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
45	19, 43, 44	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
46		1nn 12218	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ ℕ)
48		ffvelcdm 7058	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
49	4, 46, 48	sylancl 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
50	49, 21	eleqtrdi 2871	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
51	50	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
52		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ_≥‘(𝐺‘1)))
53		elfzuzb 13520	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))))
54	51, 52, 53	sylanbrc 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
55	5	ffnd 6688	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺 Fn ℕ)
56		elpreima 7035	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ (^◡𝐺 “ (𝑀...𝑁)))
59	58	ne0d 4294	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
60		nnssre 12211	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℝ
61	6, 60	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)
62		fisupcl 9413	. . . . . . . . . . . . 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 1390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)))
64	6, 63	sseldd 3937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
65	64	nnzd 12591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ)
66		elfz5 13518	. . . . . . . . . 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 606	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
68		elpreima 7035	. . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))
71		elfzle2 13530	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
72	70, 71	simpl2im 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
73	72	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
74		uzssz 12857	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
75	21, 74	eqsstri 3982	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 ⊆ ℤ
76		zssre 12572	. . . . . . . . . . . . . . . 16 ⊢ ℤ ⊆ ℝ
77	75, 76	sstri 3945	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℝ
78	5	ffvelcdmda 7061	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍)
79	77, 78	sselid 3934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
80	5, 64	ffvelcdmd 7062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
81	80	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
82	77, 81	sselid 3934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ)
83		eluzelz 12846	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐺‘1)) → 𝑁 ∈ ℤ)
84	83	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
85	76, 84	sselid 3934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
86		letr 11274	. . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . 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 704	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺‘𝑥) ≤ 𝑁))
89	30	ad2antrr 736	. . . . . . . . . . . . 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 11223	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
92	90, 91	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ^*)
93		imassrn 6057	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 “ ℕ) ⊆ ran 𝐺
94	4	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
95	94	frnd 6696	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ 𝑍)
96	93, 95	sstrid 3947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍)
97	96, 77	sstrdi 3948	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
98	97, 91	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ^*)
99		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
100	64	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
101		leisorel 14470	. . . . . . . . . . . . 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 1397	. . . . . . . . . . . 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 2871	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ_≥‘𝑀))
104		elfz5 13518	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑥) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
105	103, 84, 104	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
106	88, 102, 105	3imtr4d 296	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → (𝐺‘𝑥) ∈ (𝑀...𝑁)))
107		elpreima 7035	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn ℕ → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...𝑁))))
108	107	baibd 547	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn ℕ ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
109	55, 108	sylan 589	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
110	106, 109	sylibrd 261	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
111		fimaxre2 12134	. . . . . . . . . . . . 13 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
112	61, 45, 111	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
113		suprub 12150	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
116	115	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
117	110, 116	impbid 214	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
118	67, 117	bitrd 281	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
119	118	ex 416	. . . . . . 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 381	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
121	120	eqrdv 2759	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (^◡𝐺 “ (𝑀...𝑁)))
122	121	fveq2d 6867	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (♯‘(^◡𝐺 “ (𝑀...𝑁))))
123	64	nnnn0d 12539	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ₀)
124		hashfz1 14356	. . . . 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 14357	. . . . . 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 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ((♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) ↔ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
128	43, 127	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
129	122, 125, 128	3eqtr3d 2804	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
130	129	oveq2d 7408	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))))
131	130, 121	eqtr3d 2798	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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 class class class wbr 5099 Or wor 5552 ^◡ccnv 5644 ran crn 5646 ↾ cres 5647 “ cima 5648 Fun wfun 6511 Fn wfn 6512 ⟶wf 6513 –1-1→wf1 6514 –1-1-onto→wf1o 6516 ‘cfv 6517 Isom wiso 6518 (class class class)co 7392 ≈ cen 8920 Fincfn 8923 supcsup 9383 ℝcr 11069 1c1 11071 + caddc 11073 ℝ^*cxr 11212 < clt 11213 ≤ cle 11214 ℕcn 12207 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ...cfz 13509 ♯chash 14340

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-hash 14341

This theorem is referenced by: isercolllem3 15677

W3C validator