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Theorem isercolllem2 14338
 Description: Lemma for isercoll 14340. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
isercoll.z 𝑍 = (ℤ𝑀)
isercoll.m (𝜑𝑀 ∈ ℤ)
isercoll.g (𝜑𝐺:ℕ⟶𝑍)
isercoll.i ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
Assertion
Ref Expression
isercolllem2 ((𝜑𝑁 ∈ (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.
StepHypRef Expression
1 elfznn 12320 . . . . . . . 8 (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈ ℕ)
21a1i 11 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈ ℕ))
3 cnvimass 5449 . . . . . . . . 9 (𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
4 isercoll.g . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶𝑍)
54adantr 481 . . . . . . . . . 10 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
6 fdm 6013 . . . . . . . . . 10 (𝐺:ℕ⟶𝑍 → dom 𝐺 = ℕ)
75, 6syl 17 . . . . . . . . 9 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → dom 𝐺 = ℕ)
83, 7syl5sseq 3637 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
98sseld 3586 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ))
10 id 22 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
11 nnuz 11675 . . . . . . . . . . 11 ℕ = (ℤ‘1)
1210, 11syl6eleq 2708 . . . . . . . . . 10 (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ‘1))
13 ltso 10070 . . . . . . . . . . . . . 14 < Or ℝ
1413a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → < Or ℝ)
15 fzfid 12720 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin)
16 ffun 6010 . . . . . . . . . . . . . . . . 17 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
17 funimacnv 5933 . . . . . . . . . . . . . . . . 17 (Fun 𝐺 → (𝐺 “ (𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
185, 16, 173syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
19 inss1 3816 . . . . . . . . . . . . . . . 16 ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
2018, 19syl6eqss 3639 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
21 ssfi 8132 . . . . . . . . . . . . . . 15 (((𝑀...𝑁) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁)) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ∈ Fin)
2215, 20, 21syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ∈ Fin)
23 ssid 3608 . . . . . . . . . . . . . . . . . . . . 21 ℕ ⊆ ℕ
24 isercoll.z . . . . . . . . . . . . . . . . . . . . . 22 𝑍 = (ℤ𝑀)
25 isercoll.m . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑀 ∈ ℤ)
26 isercoll.i . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
2724, 25, 4, 26isercolllem1 14337 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
2823, 27mpan2 706 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
29 ffn 6007 . . . . . . . . . . . . . . . . . . . . 21 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
30 fnresdm 5963 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
31 isoeq1 6527 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
324, 29, 30, 314syl 19 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
3328, 32mpbid 222 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
34 isof1o 6533 . . . . . . . . . . . . . . . . . . 19 (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
35 f1ocnv 6111 . . . . . . . . . . . . . . . . . . 19 (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → 𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
36 f1ofun 6101 . . . . . . . . . . . . . . . . . . 19 (𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun 𝐺)
3733, 34, 35, 364syl 19 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun 𝐺)
38 df-f1 5857 . . . . . . . . . . . . . . . . . 18 (𝐺:ℕ–1-1𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun 𝐺))
394, 37, 38sylanbrc 697 . . . . . . . . . . . . . . . . 17 (𝜑𝐺:ℕ–1-1𝑍)
4039adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝐺:ℕ–1-1𝑍)
41 nnex 10978 . . . . . . . . . . . . . . . . 17 ℕ ∈ V
42 ssexg 4769 . . . . . . . . . . . . . . . . 17 (((𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V) → (𝐺 “ (𝑀...𝑁)) ∈ V)
438, 41, 42sylancl 693 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ∈ V)
44 f1imaeng 7968 . . . . . . . . . . . . . . . 16 ((𝐺:ℕ–1-1𝑍 ∧ (𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ (𝐺 “ (𝑀...𝑁)) ∈ V) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ≈ (𝐺 “ (𝑀...𝑁)))
4540, 8, 43, 44syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ≈ (𝐺 “ (𝑀...𝑁)))
4645ensymd 7959 . . . . . . . . . . . . . 14 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (𝐺 “ (𝑀...𝑁))))
47 enfii 8129 . . . . . . . . . . . . . 14 (((𝐺 “ (𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) → (𝐺 “ (𝑀...𝑁)) ∈ Fin)
4822, 46, 47syl2anc 692 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ∈ Fin)
49 1nn 10983 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
5049a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ ℕ)
51 ffvelrn 6318 . . . . . . . . . . . . . . . . . . 19 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
524, 49, 51sylancl 693 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺‘1) ∈ 𝑍)
5352, 24syl6eleq 2708 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
5453adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ𝑀))
55 simpr 477 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝑁 ∈ (ℤ‘(𝐺‘1)))
56 elfzuzb 12286 . . . . . . . . . . . . . . . 16 ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ‘(𝐺‘1))))
5754, 55, 56sylanbrc 697 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
585, 29syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝐺 Fn ℕ)
59 elpreima 6298 . . . . . . . . . . . . . . . 16 (𝐺 Fn ℕ → (1 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
6058, 59syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
6150, 57, 60mpbir2and 956 . . . . . . . . . . . . . 14 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ (𝐺 “ (𝑀...𝑁)))
62 ne0i 3902 . . . . . . . . . . . . . 14 (1 ∈ (𝐺 “ (𝑀...𝑁)) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
6361, 62syl 17 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
64 nnssre 10976 . . . . . . . . . . . . . 14 ℕ ⊆ ℝ
658, 64syl6ss 3599 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ⊆ ℝ)
66 fisupcl 8327 . . . . . . . . . . . . 13 (( < Or ℝ ∧ ((𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ (𝐺 “ (𝑀...𝑁)) ⊆ ℝ)) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (𝐺 “ (𝑀...𝑁)))
6714, 48, 63, 65, 66syl13anc 1325 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (𝐺 “ (𝑀...𝑁)))
688, 67sseldd 3588 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
6968nnzd 11433 . . . . . . . . . 10 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ)
70 elfz5 12284 . . . . . . . . . 10 ((𝑥 ∈ (ℤ‘1) ∧ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ) → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < )))
7112, 69, 70syl2anr 495 . . . . . . . . 9 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < )))
72 elpreima 6298 . . . . . . . . . . . . . . . . . 18 (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))))
7358, 72syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑁 ∈ (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))))
7467, 73mpbid 222 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))
7574simprd 479 . . . . . . . . . . . . . . 15 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁))
76 elfzle2 12295 . . . . . . . . . . . . . . 15 ((𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
7775, 76syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
7877adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁)
79 uzssz 11659 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) ⊆ ℤ
8024, 79eqsstri 3619 . . . . . . . . . . . . . . . 16 𝑍 ⊆ ℤ
81 zssre 11336 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℝ
8280, 81sstri 3596 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℝ
835ffvelrnda 6320 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ 𝑍)
8482, 83sseldi 3585 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ ℝ)
855, 68ffvelrnd 6321 . . . . . . . . . . . . . . . 16 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
8685adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍)
8782, 86sseldi 3585 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ)
88 eluzelz 11649 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘(𝐺‘1)) → 𝑁 ∈ ℤ)
8988ad2antlr 762 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
9081, 89sseldi 3585 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
91 letr 10083 . . . . . . . . . . . . . 14 (((𝐺𝑥) ∈ ℝ ∧ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝐺𝑥) ≤ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺𝑥) ≤ 𝑁))
9284, 87, 90, 91syl3anc 1323 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (((𝐺𝑥) ≤ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺𝑥) ≤ 𝑁))
9378, 92mpan2d 709 . . . . . . . . . . . 12 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ≤ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺𝑥) ≤ 𝑁))
9433ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
9564a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ)
96 ressxr 10035 . . . . . . . . . . . . . 14 ℝ ⊆ ℝ*
9795, 96syl6ss 3599 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ*)
98 imassrn 5441 . . . . . . . . . . . . . . . 16 (𝐺 “ ℕ) ⊆ ran 𝐺
994ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
100 frn 6015 . . . . . . . . . . . . . . . . 17 (𝐺:ℕ⟶𝑍 → ran 𝐺𝑍)
10199, 100syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺𝑍)
10298, 101syl5ss 3598 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍)
103102, 82syl6ss 3599 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
104103, 96syl6ss 3599 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ*)
105 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
10668adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)
107 leisorel 13190 . . . . . . . . . . . . 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, < ) ↔ (𝐺𝑥) ≤ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))))
10894, 97, 104, 105, 106, 107syl122anc 1332 . . . . . . . . . . . 12 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ (𝐺𝑥) ≤ (𝐺‘sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))))
10983, 24syl6eleq 2708 . . . . . . . . . . . . 13 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ (ℤ𝑀))
110 elfz5 12284 . . . . . . . . . . . . 13 (((𝐺𝑥) ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺𝑥) ∈ (𝑀...𝑁) ↔ (𝐺𝑥) ≤ 𝑁))
111109, 89, 110syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...𝑁) ↔ (𝐺𝑥) ≤ 𝑁))
11293, 108, 1113imtr4d 283 . . . . . . . . . . 11 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) → (𝐺𝑥) ∈ (𝑀...𝑁)))
113 elpreima 6298 . . . . . . . . . . . . 13 (𝐺 Fn ℕ → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...𝑁))))
114113baibd 947 . . . . . . . . . . . 12 ((𝐺 Fn ℕ ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (𝐺𝑥) ∈ (𝑀...𝑁)))
11558, 114sylan 488 . . . . . . . . . . 11 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (𝐺𝑥) ∈ (𝑀...𝑁)))
116112, 115sylibrd 249 . . . . . . . . . 10 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) → 𝑥 ∈ (𝐺 “ (𝑀...𝑁))))
117 fimaxre2 10921 . . . . . . . . . . . . 13 (((𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐺 “ (𝑀...𝑁))𝑦𝑥)
11865, 48, 117syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐺 “ (𝑀...𝑁))𝑦𝑥)
119 suprub 10936 . . . . . . . . . . . . 13 ((((𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐺 “ (𝑀...𝑁))𝑦𝑥) ∧ 𝑥 ∈ (𝐺 “ (𝑀...𝑁))) → 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))
120119ex 450 . . . . . . . . . . . 12 (((𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐺 “ (𝑀...𝑁))𝑦𝑥) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < )))
12165, 63, 118, 120syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < )))
122121adantr 481 . . . . . . . . . 10 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < )))
123116, 122impbid 202 . . . . . . . . 9 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ 𝑥 ∈ (𝐺 “ (𝑀...𝑁))))
12471, 123bitrd 268 . . . . . . . 8 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (𝐺 “ (𝑀...𝑁))))
125124ex 450 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑥 ∈ ℕ → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (𝐺 “ (𝑀...𝑁)))))
1262, 9, 125pm5.21ndd 369 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑥 ∈ (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (𝐺 “ (𝑀...𝑁))))
127126eqrdv 2619 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) = (𝐺 “ (𝑀...𝑁)))
128127fveq2d 6157 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (#‘(1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (#‘(𝐺 “ (𝑀...𝑁))))
12968nnnn0d 11303 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ0)
130 hashfz1 13082 . . . . 5 (sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ0 → (#‘(1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))
131129, 130syl 17 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (#‘(1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((𝐺 “ (𝑀...𝑁)), ℝ, < ))
132 hashen 13083 . . . . . 6 (((𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑁))) ∈ Fin) → ((#‘(𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
13348, 22, 132syl2anc 692 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((#‘(𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
13446, 133mpbird 247 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (#‘(𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))
135128, 131, 1343eqtr3d 2663 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → sup((𝐺 “ (𝑀...𝑁)), ℝ, < ) = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))
136135oveq2d 6626 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...sup((𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))))
137136, 127eqtr3d 2657 1 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) = (𝐺 “ (𝑀...𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896   class class class wbr 4618   Or wor 4999  ◡ccnv 5078  dom cdm 5079  ran crn 5080   ↾ cres 5081   “ cima 5082  Fun wfun 5846   Fn wfn 5847  ⟶wf 5848  –1-1→wf1 5849  –1-1-onto→wf1o 5851  ‘cfv 5852   Isom wiso 5853  (class class class)co 6610   ≈ cen 7904  Fincfn 7907  supcsup 8298  ℝcr 9887  1c1 9889   + caddc 9891  ℝ*cxr 10025   < clt 10026   ≤ cle 10027  ℕcn 10972  ℕ0cn0 11244  ℤcz 11329  ℤ≥cuz 11639  ...cfz 12276  #chash 13065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066 This theorem is referenced by:  isercolllem3  14339
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