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Theorem isercolllem2 15628

Description: Lemma for isercoll 15630. (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 13507	. . . . . . . 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 6047	. . . . . . . . 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 6687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
7	6	sseld 3920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ))
8		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
9		nnuz 12827	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2846	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ_≥‘1))
11		ltso 11226	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → < Or ℝ)
13		fzfid 13935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin)
14		ffun 6671	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
15		funimacnv 6579	. . . . . . . . . . . . . . . . 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 4177	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
18	16, 17	eqsstrdi 3966	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
19	13, 18	ssfid 9179	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin)
20		ssid 3944	. . . . . . . . . . . . . . . . . . . . 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 15627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25	20, 24	mpan2 692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26		ffn 6668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
27		fnresdm 6617	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28		isoeq1 7272	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
31		isof1o 7278	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32		f1ocnv 6792	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33		f1ofun 6782	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ^◡𝐺)
34	30, 31, 32, 33	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ^◡𝐺)
35		df-f1 6503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ^◡𝐺))
36	4, 34, 35	sylanbrc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍)
37	36	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 𝐺:ℕ–1-1→𝑍)
38		nnex 12180	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
39		ssexg 5264	. . . . . . . . . . . . . . . . 17 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
40	6, 38, 39	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ V)
41		f1imaeng 8961	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ≈ (^◡𝐺 “ (𝑀...𝑁)))
43	42	ensymd 8952	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))
44		enfii 9120	. . . . . . . . . . . . . 14 ⊢ (((𝐺 “ (^◡𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
45	19, 43, 44	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin)
46		1nn 12185	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ ℕ)
48		ffvelcdm 7033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
49	4, 46, 48	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
50	49, 21	eleqtrdi 2846	. . . . . . . . . . . . . . . . 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 13472	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))))
54	51, 52, 53	sylanbrc 584	. . . . . . . . . . . . . . 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 7010	. . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → 1 ∈ (^◡𝐺 “ (𝑀...𝑁)))
59	58	ne0d 4282	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
60		nnssre 12178	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℝ
61	6, 60	sstrdi 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)
62		fisupcl 9383	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (^◡𝐺 “ (𝑀...𝑁)))
64	6, 63	sseldd 3922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
65	64	nnzd 12550	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ)
66		elfz5 13470	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )))
68		elpreima 7010	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))
71		elfzle2 13482	. . . . . . . . . . . . . . 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 12809	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
75	21, 74	eqsstri 3968	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 ⊆ ℤ
76		zssre 12531	. . . . . . . . . . . . . . . 16 ⊢ ℤ ⊆ ℝ
77	75, 76	sstri 3931	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℝ
78	5	ffvelcdmda 7036	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍)
79	77, 78	sselid 3919	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
80	5, 64	ffvelcdmd 7037	. . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ)
83		eluzelz 12798	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐺‘1)) → 𝑁 ∈ ℤ)
84	83	ad2antlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
85	76, 84	sselid 3919	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
86		letr 11240	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ≤ (𝐺‘sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺‘𝑥) ≤ 𝑁))
89	30	ad2antrr 727	. . . . . . . . . . . . 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 11189	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
92	90, 91	sstrdi 3934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ^*)
93		imassrn 6036	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 “ ℕ) ⊆ ran 𝐺
94	4	ad2antrr 727	. . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍)
97	96, 77	sstrdi 3934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
98	97, 91	sstrdi 3934	. . . . . . . . . . . . 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 14422	. . . . . . . . . . . . 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 1382	. . . . . . . . . . . 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 2846	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ_≥‘𝑀))
104		elfz5 13470	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑥) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
105	103, 84, 104	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁))
106	88, 102, 105	3imtr4d 294	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → (𝐺‘𝑥) ∈ (𝑀...𝑁)))
107		elpreima 7010	. . . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁)))
110	106, 109	sylibrd 259	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
111		fimaxre2 12101	. . . . . . . . . . . . 13 ⊢ (((^◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (^◡𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
112	61, 45, 111	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (^◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥)
113		suprub 12117	. . . . . . . . . . . . 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 1374	. . . . . . . . . . 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 212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ 𝑥 ∈ (^◡𝐺 “ (𝑀...𝑁))))
118	67, 117	bitrd 279	. . . . . . . 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 379	. . . . . 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 6844	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (♯‘(^◡𝐺 “ (𝑀...𝑁))))
123	64	nnnn0d 12498	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ₀)
124		hashfz1 14308	. . . . 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 14309	. . . . . 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 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → ((♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))) ↔ (^◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
128	43, 127	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (♯‘(^◡𝐺 “ (𝑀...𝑁))) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
129	122, 125, 128	3eqtr3d 2779	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < ) = (♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁)))))
130	129	oveq2d 7383	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ_≥‘(𝐺‘1))) → (1...sup((^◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(♯‘(𝐺 “ (^◡𝐺 “ (𝑀...𝑁))))))
131	130, 121	eqtr3d 2773	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 Or wor 5538 ^◡ccnv 5630 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6492 Fn wfn 6493 ⟶wf 6494 –1-1→wf1 6495 –1-1-onto→wf1o 6497 ‘cfv 6498 Isom wiso 6499 (class class class)co 7367 ≈ cen 8890 Fincfn 8893 supcsup 9353 ℝcr 11037 1c1 11039 + caddc 11041 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ♯chash 14292

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-hash 14293

This theorem is referenced by: isercolllem3 15629

W3C validator