MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqcoll Structured version   Visualization version   GIF version

Theorem seqcoll 13057
Description: The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
Hypotheses
Ref Expression
seqcoll.1 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
seqcoll.1b ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
seqcoll.c ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
seqcoll.a (𝜑𝑍𝑆)
seqcoll.2 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
seqcoll.3 (𝜑𝑁 ∈ (1...(#‘𝐴)))
seqcoll.4 (𝜑𝐴 ⊆ (ℤ𝑀))
seqcoll.5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
seqcoll.6 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
seqcoll.7 ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
Assertion
Ref Expression
seqcoll (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑛,𝐻   𝑘,𝑀,𝑛   + ,𝑘,𝑛   𝜑,𝑘,𝑛   𝑆,𝑘,𝑛   𝑘,𝑍
Allowed substitution hints:   𝐻(𝑘)   𝑁(𝑘,𝑛)   𝑍(𝑛)

Proof of Theorem seqcoll
Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqcoll.3 . 2 (𝜑𝑁 ∈ (1...(#‘𝐴)))
2 elfznn 12196 . . . 4 (𝑁 ∈ (1...(#‘𝐴)) → 𝑁 ∈ ℕ)
31, 2syl 17 . . 3 (𝜑𝑁 ∈ ℕ)
4 eleq1 2675 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 1 ∈ (1...(#‘𝐴))))
5 fveq2 6088 . . . . . . . 8 (𝑦 = 1 → (𝐺𝑦) = (𝐺‘1))
65fveq2d 6092 . . . . . . 7 (𝑦 = 1 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺‘1)))
7 fveq2 6088 . . . . . . 7 (𝑦 = 1 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘1))
86, 7eqeq12d 2624 . . . . . 6 (𝑦 = 1 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1)))
94, 8imbi12d 332 . . . . 5 (𝑦 = 1 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (1 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1))))
109imbi2d 328 . . . 4 (𝑦 = 1 → ((𝜑 → (𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦))) ↔ (𝜑 → (1 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1)))))
11 eleq1 2675 . . . . . 6 (𝑦 = 𝑚 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 𝑚 ∈ (1...(#‘𝐴))))
12 fveq2 6088 . . . . . . . 8 (𝑦 = 𝑚 → (𝐺𝑦) = (𝐺𝑚))
1312fveq2d 6092 . . . . . . 7 (𝑦 = 𝑚 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺𝑚)))
14 fveq2 6088 . . . . . . 7 (𝑦 = 𝑚 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘𝑚))
1513, 14eqeq12d 2624 . . . . . 6 (𝑦 = 𝑚 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)))
1611, 15imbi12d 332 . . . . 5 (𝑦 = 𝑚 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚))))
1716imbi2d 328 . . . 4 (𝑦 = 𝑚 → ((𝜑 → (𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦))) ↔ (𝜑 → (𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)))))
18 eleq1 2675 . . . . . 6 (𝑦 = (𝑚 + 1) → (𝑦 ∈ (1...(#‘𝐴)) ↔ (𝑚 + 1) ∈ (1...(#‘𝐴))))
19 fveq2 6088 . . . . . . . 8 (𝑦 = (𝑚 + 1) → (𝐺𝑦) = (𝐺‘(𝑚 + 1)))
2019fveq2d 6092 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))))
21 fveq2 6088 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘(𝑚 + 1)))
2220, 21eqeq12d 2624 . . . . . 6 (𝑦 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))
2318, 22imbi12d 332 . . . . 5 (𝑦 = (𝑚 + 1) → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1)))))
2423imbi2d 328 . . . 4 (𝑦 = (𝑚 + 1) → ((𝜑 → (𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦))) ↔ (𝜑 → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))))
25 eleq1 2675 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 𝑁 ∈ (1...(#‘𝐴))))
26 fveq2 6088 . . . . . . . 8 (𝑦 = 𝑁 → (𝐺𝑦) = (𝐺𝑁))
2726fveq2d 6092 . . . . . . 7 (𝑦 = 𝑁 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺𝑁)))
28 fveq2 6088 . . . . . . 7 (𝑦 = 𝑁 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘𝑁))
2927, 28eqeq12d 2624 . . . . . 6 (𝑦 = 𝑁 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁)))
3025, 29imbi12d 332 . . . . 5 (𝑦 = 𝑁 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))))
3130imbi2d 328 . . . 4 (𝑦 = 𝑁 → ((𝜑 → (𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦))) ↔ (𝜑 → (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁)))))
32 seqcoll.1 . . . . . . . . 9 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
33 seqcoll.a . . . . . . . . 9 (𝜑𝑍𝑆)
34 seqcoll.4 . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℤ𝑀))
35 seqcoll.2 . . . . . . . . . . . . 13 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
36 isof1o 6451 . . . . . . . . . . . . 13 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
3735, 36syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
38 f1of 6035 . . . . . . . . . . . 12 (𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝐺:(1...(#‘𝐴))⟶𝐴)
3937, 38syl 17 . . . . . . . . . . 11 (𝜑𝐺:(1...(#‘𝐴))⟶𝐴)
40 elfzuz2 12172 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(#‘𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
411, 40syl 17 . . . . . . . . . . . 12 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
42 eluzfz1 12174 . . . . . . . . . . . 12 ((#‘𝐴) ∈ (ℤ‘1) → 1 ∈ (1...(#‘𝐴)))
4341, 42syl 17 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...(#‘𝐴)))
4439, 43ffvelrnd 6253 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ 𝐴)
4534, 44sseldd 3568 . . . . . . . . 9 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
46 eluzle 11532 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ (ℤ‘1) → 1 ≤ (#‘𝐴))
4741, 46syl 17 . . . . . . . . . . . 12 (𝜑 → 1 ≤ (#‘𝐴))
48 elfzelz 12168 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...(#‘𝐴)) → 𝑘 ∈ ℤ)
4948ssriv 3571 . . . . . . . . . . . . . . . 16 (1...(#‘𝐴)) ⊆ ℤ
50 zssre 11217 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℝ
5149, 50sstri 3576 . . . . . . . . . . . . . . 15 (1...(#‘𝐴)) ⊆ ℝ
5251a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (1...(#‘𝐴)) ⊆ ℝ)
53 ressxr 9939 . . . . . . . . . . . . . 14 ℝ ⊆ ℝ*
5452, 53syl6ss 3579 . . . . . . . . . . . . 13 (𝜑 → (1...(#‘𝐴)) ⊆ ℝ*)
55 eluzelre 11530 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℝ)
5655ssriv 3571 . . . . . . . . . . . . . . 15 (ℤ𝑀) ⊆ ℝ
5734, 56syl6ss 3579 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
5857, 53syl6ss 3579 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℝ*)
59 eluzfz2 12175 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
6041, 59syl 17 . . . . . . . . . . . . 13 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
61 leisorel 13053 . . . . . . . . . . . . 13 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((1...(#‘𝐴)) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ (1 ∈ (1...(#‘𝐴)) ∧ (#‘𝐴) ∈ (1...(#‘𝐴)))) → (1 ≤ (#‘𝐴) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
6235, 54, 58, 43, 60, 61syl122anc 1326 . . . . . . . . . . . 12 (𝜑 → (1 ≤ (#‘𝐴) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
6347, 62mpbid 220 . . . . . . . . . . 11 (𝜑 → (𝐺‘1) ≤ (𝐺‘(#‘𝐴)))
6439, 60ffvelrnd 6253 . . . . . . . . . . . . . 14 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
6534, 64sseldd 3568 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
66 eluzelz 11529 . . . . . . . . . . . . 13 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(#‘𝐴)) ∈ ℤ)
6765, 66syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℤ)
68 elfz5 12160 . . . . . . . . . . . 12 (((𝐺‘1) ∈ (ℤ𝑀) ∧ (𝐺‘(#‘𝐴)) ∈ ℤ) → ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
6945, 67, 68syl2anc 690 . . . . . . . . . . 11 (𝜑 → ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
7063, 69mpbird 245 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))))
71 fveq2 6088 . . . . . . . . . . . . 13 (𝑘 = (𝐺‘1) → (𝐹𝑘) = (𝐹‘(𝐺‘1)))
7271eleq1d 2671 . . . . . . . . . . . 12 (𝑘 = (𝐺‘1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝐺‘1)) ∈ 𝑆))
7372imbi2d 328 . . . . . . . . . . 11 (𝑘 = (𝐺‘1) → ((𝜑 → (𝐹𝑘) ∈ 𝑆) ↔ (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)))
74 seqcoll.5 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
7574expcom 449 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))) → (𝜑 → (𝐹𝑘) ∈ 𝑆))
7673, 75vtoclga 3244 . . . . . . . . . 10 ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) → (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆))
7770, 76mpcom 37 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)
78 eluzelz 11529 . . . . . . . . . . . . . . . . . 18 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
7945, 78syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺‘1) ∈ ℤ)
80 peano2zm 11253 . . . . . . . . . . . . . . . . 17 ((𝐺‘1) ∈ ℤ → ((𝐺‘1) − 1) ∈ ℤ)
8179, 80syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐺‘1) − 1) ∈ ℤ)
8281zred 11314 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ∈ ℝ)
8379zred 11314 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘1) ∈ ℝ)
8467zred 11314 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
8583lem1d 10806 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘1))
8682, 83, 84, 85, 63letrd 10045 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴)))
87 eluz 11533 . . . . . . . . . . . . . . 15 ((((𝐺‘1) − 1) ∈ ℤ ∧ (𝐺‘(#‘𝐴)) ∈ ℤ) → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴))))
8881, 67, 87syl2anc 690 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴))))
8986, 88mpbird 245 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)))
90 fzss2 12207 . . . . . . . . . . . . 13 ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) → (𝑀...((𝐺‘1) − 1)) ⊆ (𝑀...(𝐺‘(#‘𝐴))))
9189, 90syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑀...((𝐺‘1) − 1)) ⊆ (𝑀...(𝐺‘(#‘𝐴))))
9291sselda 3567 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
93 eluzel2 11524 . . . . . . . . . . . . . . 15 ((𝐺‘1) ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
9445, 93syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
95 elfzm11 12235 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝐺‘1) ∈ ℤ) → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
9694, 79, 95syl2anc 690 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
97 simp3 1055 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → 𝑘 < (𝐺‘1))
98 f1ocnv 6047 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
9937, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
100 f1of 6035 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺:𝐴1-1-onto→(1...(#‘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
10199, 100syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐺:𝐴⟶(1...(#‘𝐴)))
102101ffvelrnda 6252 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
103 elfznn 12196 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℕ)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ ℕ)
105104nnge1d 10910 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → 1 ≤ (𝐺𝑘))
10635adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
10754adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (1...(#‘𝐴)) ⊆ ℝ*)
10858adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐴 ⊆ ℝ*)
10943adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 1 ∈ (1...(#‘𝐴)))
110 leisorel 13053 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((1...(#‘𝐴)) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ (1 ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → (1 ≤ (𝐺𝑘) ↔ (𝐺‘1) ≤ (𝐺‘(𝐺𝑘))))
111106, 107, 108, 109, 102, 110syl122anc 1326 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → (1 ≤ (𝐺𝑘) ↔ (𝐺‘1) ≤ (𝐺‘(𝐺𝑘))))
112105, 111mpbid 220 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ (𝐺‘(𝐺𝑘)))
113 f1ocnvfv2 6411 . . . . . . . . . . . . . . . . . . 19 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
11437, 113sylan 486 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
115112, 114breqtrd 4603 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ 𝑘)
11683adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘1) ∈ ℝ)
11757sselda 3567 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑘 ∈ ℝ)
118116, 117lenltd 10034 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → ((𝐺‘1) ≤ 𝑘 ↔ ¬ 𝑘 < (𝐺‘1)))
119115, 118mpbid 220 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ¬ 𝑘 < (𝐺‘1))
120119ex 448 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘𝐴 → ¬ 𝑘 < (𝐺‘1)))
121120con2d 127 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 < (𝐺‘1) → ¬ 𝑘𝐴))
12297, 121syl5 33 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → ¬ 𝑘𝐴))
12396, 122sylbid 228 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) → ¬ 𝑘𝐴))
124123imp 443 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → ¬ 𝑘𝐴)
12592, 124eldifd 3550 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴))
126 seqcoll.6 . . . . . . . . . 10 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
127125, 126syldan 485 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → (𝐹𝑘) = 𝑍)
12832, 33, 45, 77, 127seqid 12663 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1))) = seq(𝐺‘1)( + , 𝐹))
129128fveq1d 6090 . . . . . . 7 (𝜑 → ((seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)))
130 uzid 11534 . . . . . . . . 9 ((𝐺‘1) ∈ ℤ → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
13179, 130syl 17 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
132 fvres 6102 . . . . . . . 8 ((𝐺‘1) ∈ (ℤ‘(𝐺‘1)) → ((seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq𝑀( + , 𝐹)‘(𝐺‘1)))
133131, 132syl 17 . . . . . . 7 (𝜑 → ((seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq𝑀( + , 𝐹)‘(𝐺‘1)))
134 seq1 12631 . . . . . . . . 9 ((𝐺‘1) ∈ ℤ → (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)) = (𝐹‘(𝐺‘1)))
13579, 134syl 17 . . . . . . . 8 (𝜑 → (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)) = (𝐹‘(𝐺‘1)))
136 fveq2 6088 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐻𝑛) = (𝐻‘1))
137 fveq2 6088 . . . . . . . . . . . . 13 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
138137fveq2d 6092 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘1)))
139136, 138eqeq12d 2624 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘1) = (𝐹‘(𝐺‘1))))
140139imbi2d 328 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))))
141 seqcoll.7 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
142141expcom 449 . . . . . . . . . 10 (𝑛 ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))))
143140, 142vtoclga 3244 . . . . . . . . 9 (1 ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1))))
14443, 143mpcom 37 . . . . . . . 8 (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))
145135, 144eqtr4d 2646 . . . . . . 7 (𝜑 → (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)) = (𝐻‘1))
146129, 133, 1453eqtr3d 2651 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (𝐻‘1))
147 1z 11240 . . . . . . 7 1 ∈ ℤ
148 seq1 12631 . . . . . . 7 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
149147, 148ax-mp 5 . . . . . 6 (seq1( + , 𝐻)‘1) = (𝐻‘1)
150146, 149syl6eqr 2661 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1))
151150a1d 25 . . . 4 (𝜑 → (1 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1)))
152 simplr 787 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℕ)
153 nnuz 11555 . . . . . . . . . . 11 ℕ = (ℤ‘1)
154152, 153syl6eleq 2697 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ (ℤ‘1))
155 nnz 11232 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
156155ad2antlr 758 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℤ)
157 elfzuz3 12165 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (#‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
158157adantl 480 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
159 peano2uzr 11575 . . . . . . . . . . 11 ((𝑚 ∈ ℤ ∧ (#‘𝐴) ∈ (ℤ‘(𝑚 + 1))) → (#‘𝐴) ∈ (ℤ𝑚))
160156, 158, 159syl2anc 690 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (ℤ𝑚))
161 elfzuzb 12162 . . . . . . . . . 10 (𝑚 ∈ (1...(#‘𝐴)) ↔ (𝑚 ∈ (ℤ‘1) ∧ (#‘𝐴) ∈ (ℤ𝑚)))
162154, 160, 161sylanbrc 694 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ (1...(#‘𝐴)))
163162ex 448 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → 𝑚 ∈ (1...(#‘𝐴))))
164163imim1d 79 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚))))
165 oveq1 6534 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq1( + , 𝐻)‘𝑚) + (𝐻‘(𝑚 + 1))))
166 simpll 785 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝜑)
167 seqcoll.1b . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
168166, 167sylan 486 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
16934ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐴 ⊆ (ℤ𝑀))
17039ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐺:(1...(#‘𝐴))⟶𝐴)
171170, 162ffvelrnd 6253 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ 𝐴)
172169, 171sseldd 3568 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ (ℤ𝑀))
173 nnre 10874 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
174173ad2antlr 758 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℝ)
175174ltp1d 10803 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 < (𝑚 + 1))
17635ad2antrr 757 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
177 simpr 475 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 + 1) ∈ (1...(#‘𝐴)))
178 isorel 6454 . . . . . . . . . . . . . . . . . 18 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(#‘𝐴)) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴)))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
179176, 162, 177, 178syl12anc 1315 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
180175, 179mpbid 220 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) < (𝐺‘(𝑚 + 1)))
181 eluzelz 11529 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑚) ∈ (ℤ𝑀) → (𝐺𝑚) ∈ ℤ)
182172, 181syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ ℤ)
183170, 177ffvelrnd 6253 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ 𝐴)
184169, 183sseldd 3568 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ𝑀))
185 eluzelz 11529 . . . . . . . . . . . . . . . . . 18 ((𝐺‘(𝑚 + 1)) ∈ (ℤ𝑀) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
186184, 185syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
187 zltlem1 11263 . . . . . . . . . . . . . . . . 17 (((𝐺𝑚) ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
188182, 186, 187syl2anc 690 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
189180, 188mpbid 220 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1))
190 peano2zm 11253 . . . . . . . . . . . . . . . . 17 ((𝐺‘(𝑚 + 1)) ∈ ℤ → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ)
191186, 190syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ)
192 eluz 11533 . . . . . . . . . . . . . . . 16 (((𝐺𝑚) ∈ ℤ ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
193182, 191, 192syl2anc 690 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
194189, 193mpbird 245 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)))
195191zred 11314 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℝ)
196186zred 11314 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℝ)
19784ad2antrr 757 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ ℝ)
198196lem1d 10806 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(𝑚 + 1)))
199 elfzle2 12171 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (𝑚 + 1) ≤ (#‘𝐴))
200199adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 + 1) ≤ (#‘𝐴))
20154ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (1...(#‘𝐴)) ⊆ ℝ*)
20258ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐴 ⊆ ℝ*)
20360ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (1...(#‘𝐴)))
204 leisorel 13053 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((1...(#‘𝐴)) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ (#‘𝐴) ∈ (1...(#‘𝐴)))) → ((𝑚 + 1) ≤ (#‘𝐴) ↔ (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(#‘𝐴))))
205176, 201, 202, 177, 203, 204syl122anc 1326 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝑚 + 1) ≤ (#‘𝐴) ↔ (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(#‘𝐴))))
206200, 205mpbid 220 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(#‘𝐴)))
207195, 196, 197, 198, 206letrd 10045 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(#‘𝐴)))
20867ad2antrr 757 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ ℤ)
209 eluz 11533 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ ∧ (𝐺‘(#‘𝐴)) ∈ ℤ) → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ↔ ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(#‘𝐴))))
210191, 208, 209syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ↔ ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(#‘𝐴))))
211207, 210mpbird 245 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)))
212 uztrn 11536 . . . . . . . . . . . . . . . . . . 19 (((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘(𝐺𝑚)))
213211, 194, 212syl2anc 690 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘(𝐺𝑚)))
214 fzss2 12207 . . . . . . . . . . . . . . . . . 18 ((𝐺‘(#‘𝐴)) ∈ (ℤ‘(𝐺𝑚)) → (𝑀...(𝐺𝑚)) ⊆ (𝑀...(𝐺‘(#‘𝐴))))
215213, 214syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑀...(𝐺𝑚)) ⊆ (𝑀...(𝐺‘(#‘𝐴))))
216215sselda 3567 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺𝑚))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
217166, 74sylan 486 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
218216, 217syldan 485 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺𝑚))) → (𝐹𝑘) ∈ 𝑆)
219 seqcoll.c . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
220166, 219sylan 486 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
221172, 218, 220seqcl 12638 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) ∈ 𝑆)
222 simplll 793 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝜑)
223 elfzuz 12164 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)))
224 peano2uz 11573 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑚) ∈ (ℤ𝑀) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
225172, 224syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
226 uztrn 11536 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)) ∧ ((𝐺𝑚) + 1) ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
227223, 225, 226syl2anr 493 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (ℤ𝑀))
228 elfzuz3 12165 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘))
229 uztrn 11536 . . . . . . . . . . . . . . . . . 18 (((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘)) → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘))
230211, 228, 229syl2an 492 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘))
231 elfzuzb 12162 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝑘 ∈ (ℤ𝑀) ∧ (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘)))
232227, 230, 231sylanbrc 694 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
233 elfzle1 12170 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺𝑚) + 1) ≤ 𝑘)
234 elfzle2 12171 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))
235233, 234jca 552 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
236155ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ ℤ)
237101ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
238 simprr 791 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘𝐴)
239237, 238ffvelrnd 6253 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
240 elfzelz 12168 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
242 btwnnz 11285 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ 𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ)
2432423expib 1259 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℤ → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ))
244243con2d 127 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℤ → ((𝐺𝑘) ∈ ℤ → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1))))
245236, 241, 244sylc 62 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)))
24635ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
247162adantrr 748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ (1...(#‘𝐴)))
248 isorel 6454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
249246, 247, 239, 248syl12anc 1315 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
25037ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
251250, 238, 113syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
252251breq2d 4589 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < (𝐺‘(𝐺𝑘)) ↔ (𝐺𝑚) < 𝑘))
253182adantrr 748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑚) ∈ ℤ)
25434ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐴 ⊆ (ℤ𝑀))
255254, 238sseldd 3568 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ (ℤ𝑀))
256 eluzelz 11529 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
257255, 256syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ ℤ)
258 zltp1le 11260 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺𝑚) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
259253, 257, 258syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
260249, 252, 2593bitrd 292 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
261177adantrr 748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 + 1) ∈ (1...(#‘𝐴)))
262 isorel 6454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((𝐺𝑘) ∈ (1...(#‘𝐴)) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴)))) → ((𝐺𝑘) < (𝑚 + 1) ↔ (𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1))))
263246, 239, 261, 262syl12anc 1315 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ (𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1))))
264251breq1d 4587 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1)) ↔ 𝑘 < (𝐺‘(𝑚 + 1))))
265186adantrr 748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
266 zltlem1 11263 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
267257, 265, 266syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
268263, 264, 2673bitrd 292 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
269260, 268anbi12d 742 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) ↔ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
270245, 269mtbid 312 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
271270expr 640 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑘𝐴 → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
272271con2d 127 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)) → ¬ 𝑘𝐴))
273235, 272syl5 33 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ¬ 𝑘𝐴))
274273imp 443 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → ¬ 𝑘𝐴)
275232, 274eldifd 3550 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴))
276222, 275, 126syl2anc 690 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐹𝑘) = 𝑍)
277168, 172, 194, 221, 276seqid2 12664 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq𝑀( + , 𝐹)‘((𝐺‘(𝑚 + 1)) − 1)))
278277oveq1d 6542 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐹‘(𝐺‘(𝑚 + 1)))) = ((seq𝑀( + , 𝐹)‘((𝐺‘(𝑚 + 1)) − 1)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
279 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐻𝑛) = (𝐻‘(𝑚 + 1)))
280 fveq2 6088 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
281280fveq2d 6092 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘(𝑚 + 1))))
282279, 281eqeq12d 2624 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑚 + 1) → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
283282imbi2d 328 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))))
284283, 142vtoclga 3244 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
285284impcom 444 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
286285adantlr 746 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
287286oveq2d 6543 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
28894ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑀 ∈ ℤ)
289186zcnd 11315 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℂ)
290 ax-1cn 9850 . . . . . . . . . . . . . . 15 1 ∈ ℂ
291 npcan 10141 . . . . . . . . . . . . . . 15 (((𝐺‘(𝑚 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
292289, 290, 291sylancl 692 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
293 uztrn 11536 . . . . . . . . . . . . . . . 16 ((((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ∧ (𝐺𝑚) ∈ (ℤ𝑀)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
294194, 172, 293syl2anc 690 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
295 eluzp1p1 11545 . . . . . . . . . . . . . . 15 (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀) → (((𝐺‘(𝑚 + 1)) − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
296294, 295syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
297292, 296eqeltrrd 2688 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ‘(𝑀 + 1)))
298 seqm1 12635 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘((𝐺‘(𝑚 + 1)) − 1)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
299288, 297, 298syl2anc 690 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘((𝐺‘(𝑚 + 1)) − 1)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
300278, 287, 2993eqtr4rd 2654 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))))
301 seqp1 12633 . . . . . . . . . . . 12 (𝑚 ∈ (ℤ‘1) → (seq1( + , 𝐻)‘(𝑚 + 1)) = ((seq1( + , 𝐻)‘𝑚) + (𝐻‘(𝑚 + 1))))
302154, 301syl 17 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq1( + , 𝐻)‘(𝑚 + 1)) = ((seq1( + , 𝐻)‘𝑚) + (𝐻‘(𝑚 + 1))))
303300, 302eqeq12d 2624 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1)) ↔ ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq1( + , 𝐻)‘𝑚) + (𝐻‘(𝑚 + 1)))))
304165, 303syl5ibr 234 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))
305304ex 448 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1)))))
306305a2d 29 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1)))))
307164, 306syld 45 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1)))))
308307expcom 449 . . . . 5 (𝑚 ∈ ℕ → (𝜑 → ((𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))))
309308a2d 29 . . . 4 (𝑚 ∈ ℕ → ((𝜑 → (𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚))) → (𝜑 → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))))
31010, 17, 24, 31, 151, 309nnind 10885 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))))
3113, 310mpcom 37 . 2 (𝜑 → (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁)))
3121, 311mpd 15 1 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cdif 3536  wss 3539   class class class wbr 4577  ccnv 5027  cres 5030  wf 5786  1-1-ontowf1o 5789  cfv 5790   Isom wiso 5791  (class class class)co 6527  cc 9790  cr 9791  1c1 9793   + caddc 9795  *cxr 9929   < clt 9930  cle 9931  cmin 10117  cn 10867  cz 11210  cuz 11519  ...cfz 12152  seqcseq 12618  #chash 12934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-seq 12619
This theorem is referenced by:  seqcoll2  13058  summolem2a  14239  prodmolem2a  14449
  Copyright terms: Public domain W3C validator