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Theorem isercolllem2 15305

Description: Lemma for isercoll 15307. (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 13214	. . . . . . . 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 5978	. . . . . . . . 9 ⊢ (^◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
4		isercoll.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
6	3, 5	fssdm 6604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
7	6	sseld 3916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ))
8		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
9		nnuz 12550	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2849	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ_≥‘1))
11		ltso 10986	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → < Or ℝ)
13		fzfid 13621	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin)
14		ffun 6587	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
15		funimacnv 6499	. . . . . . . . . . . . . . . . 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 4159	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
18	16, 17	eqsstrdi 3971	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
19	13, 18	ssfid 8971	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin)
20		ssid 3939	. . . . . . . . . . . . . . . . . . . . 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 15304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25	20, 24	mpan2 687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26		ffn 6584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
27		fnresdm 6535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28		isoeq1 7168	. . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32		f1ocnv 6712	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33		f1ofun 6702	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ^◡𝐺)
34	30, 31, 32, 33	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ^◡𝐺)
35		df-f1 6423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ^◡𝐺))
36	4, 34, 35	sylanbrc 582	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍)
37	36	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ–1-1→𝑍)
38		nnex 11909	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
39		ssexg 5242	. . . . . . . . . . . . . . . . 17 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
40	6, 38, 39	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
41		f1imaeng 8755	. . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ≈ (^◡𝐺 “ (𝑀...𝑁)))
43	42	ensymd 8746	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))
44		enfii 8932	. . . . . . . . . . . . . 14 ⊢ (((𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
45	19, 43, 44	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
46		1nn 11914	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ ℕ)
48		ffvelrn 6941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
49	4, 46, 48	sylancl 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
50	49, 21	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
51	50	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ_≥‘𝑀))
52		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ_≥‘(𝐺‘1)))
53		elfzuzb 13179	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))))
54	51, 52, 53	sylanbrc 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
55	5	ffnd 6585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺 Fn ℕ)
56		elpreima 6917	. . . . . . . . . . . . . . . 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 709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ (^◡𝐺 “ (𝑀...𝑁)))
59	58	ne0d 4266	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
60		nnssre 11907	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℝ
61	6, 60	sstrdi 3929	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)
62		fisupcl 9158	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)))
64	6, 63	sseldd 3918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
65	64	nnzd 12354	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ)
66		elfz5 13177	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
68		elpreima 6917	. . . . . . . . . . . . . . . . 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 13189	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
72	70, 71	simpl2im 503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
73	72	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
74		uzssz 12532	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
75	21, 74	eqsstri 3951	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 ⊆ ℤ
76		zssre 12256	. . . . . . . . . . . . . . . 16 ⊢ ℤ ⊆ ℝ
77	75, 76	sstri 3926	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℝ
78	5	ffvelrnda 6943	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍)
79	77, 78	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
80	5, 64	ffvelrnd 6944	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
81	80	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
82	77, 81	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ)
83		eluzelz 12521	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐺‘1)) → 𝑁 ∈ ℤ)
84	83	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
85	76, 84	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
86		letr 10999	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺‘𝑥) ≤ 𝑁))
89	30	ad2antrr 722	. . . . . . . . . . . . 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 10950	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
92	90, 91	sstrdi 3929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ^*)
93		imassrn 5969	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 “ ℕ) ⊆ ran 𝐺
94	4	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
95	94	frnd 6592	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ 𝑍)
96	93, 95	sstrid 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍)
97	96, 77	sstrdi 3929	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
98	97, 91	sstrdi 3929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ^*)
99		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
100	64	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
101		leisorel 14102	. . . . . . . . . . . . 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 1377	. . . . . . . . . . . 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 2849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ_≥‘𝑀))
104		elfz5 13177	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑥) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
105	103, 84, 104	syl2anc 583	. . . . . . . . . . . 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 6917	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn ℕ → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...𝑁))))
108	107	baibd 539	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn ℕ ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
109	55, 108	sylan 579	. . . . . . . . . . 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 11850	. . . . . . . . . . . . 13 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
112	61, 45, 111	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
113		suprub 11866	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
116	115	adantr 480	. . . . . . . . . 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 412	. . . . . . 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 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (^◡𝐺 “ (𝑀...𝑁)))
122	121	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (♯‘(^◡𝐺 “ (𝑀...𝑁))))
123	64	nnnn0d 12223	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ₀)
124		hashfz1 13988	. . . . 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 13989	. . . . . 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 583	. . . . 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 2786	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
130	129	oveq2d 7271	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))))
131	130, 121	eqtr3d 2780	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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 class class class wbr 5070 Or wor 5493 ^◡ccnv 5579 ran crn 5581 ↾ cres 5582 “ cima 5583 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 Isom wiso 6419 (class class class)co 7255 ≈ cen 8688 Fincfn 8691 supcsup 9129 ℝcr 10801 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 ♯chash 13972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-hash 13973

This theorem is referenced by: isercolllem3 15306

W3C validator