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Theorem seqcoll 13286
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 12408 . . . 4 (𝑁 ∈ (1...(#‘𝐴)) → 𝑁 ∈ ℕ)
31, 2syl 17 . . 3 (𝜑𝑁 ∈ ℕ)
4 eleq1 2718 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 1 ∈ (1...(#‘𝐴))))
5 fveq2 6229 . . . . . . . 8 (𝑦 = 1 → (𝐺𝑦) = (𝐺‘1))
65fveq2d 6233 . . . . . . 7 (𝑦 = 1 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺‘1)))
7 fveq2 6229 . . . . . . 7 (𝑦 = 1 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘1))
86, 7eqeq12d 2666 . . . . . 6 (𝑦 = 1 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1)))
94, 8imbi12d 333 . . . . 5 (𝑦 = 1 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (1 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (seq1( + , 𝐻)‘1))))
109imbi2d 329 . . . 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 2718 . . . . . 6 (𝑦 = 𝑚 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 𝑚 ∈ (1...(#‘𝐴))))
12 fveq2 6229 . . . . . . . 8 (𝑦 = 𝑚 → (𝐺𝑦) = (𝐺𝑚))
1312fveq2d 6233 . . . . . . 7 (𝑦 = 𝑚 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺𝑚)))
14 fveq2 6229 . . . . . . 7 (𝑦 = 𝑚 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘𝑚))
1513, 14eqeq12d 2666 . . . . . 6 (𝑦 = 𝑚 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)))
1611, 15imbi12d 333 . . . . 5 (𝑦 = 𝑚 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚))))
1716imbi2d 329 . . . 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 2718 . . . . . 6 (𝑦 = (𝑚 + 1) → (𝑦 ∈ (1...(#‘𝐴)) ↔ (𝑚 + 1) ∈ (1...(#‘𝐴))))
19 fveq2 6229 . . . . . . . 8 (𝑦 = (𝑚 + 1) → (𝐺𝑦) = (𝐺‘(𝑚 + 1)))
2019fveq2d 6233 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))))
21 fveq2 6229 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘(𝑚 + 1)))
2220, 21eqeq12d 2666 . . . . . 6 (𝑦 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))
2318, 22imbi12d 333 . . . . 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 329 . . . 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 2718 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...(#‘𝐴)) ↔ 𝑁 ∈ (1...(#‘𝐴))))
26 fveq2 6229 . . . . . . . 8 (𝑦 = 𝑁 → (𝐺𝑦) = (𝐺𝑁))
2726fveq2d 6233 . . . . . . 7 (𝑦 = 𝑁 → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹)‘(𝐺𝑁)))
28 fveq2 6229 . . . . . . 7 (𝑦 = 𝑁 → (seq1( + , 𝐻)‘𝑦) = (seq1( + , 𝐻)‘𝑁))
2927, 28eqeq12d 2666 . . . . . 6 (𝑦 = 𝑁 → ((seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦) ↔ (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁)))
3025, 29imbi12d 333 . . . . 5 (𝑦 = 𝑁 → ((𝑦 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑦)) = (seq1( + , 𝐻)‘𝑦)) ↔ (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))))
3130imbi2d 329 . . . 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 6613 . . . . . . . . . . . . 13 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
3735, 36syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
38 f1of 6175 . . . . . . . . . . . 12 (𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝐺:(1...(#‘𝐴))⟶𝐴)
3937, 38syl 17 . . . . . . . . . . 11 (𝜑𝐺:(1...(#‘𝐴))⟶𝐴)
40 elfzuz2 12384 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(#‘𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
411, 40syl 17 . . . . . . . . . . . 12 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
42 eluzfz1 12386 . . . . . . . . . . . 12 ((#‘𝐴) ∈ (ℤ‘1) → 1 ∈ (1...(#‘𝐴)))
4341, 42syl 17 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...(#‘𝐴)))
4439, 43ffvelrnd 6400 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ 𝐴)
4534, 44sseldd 3637 . . . . . . . . 9 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
46 eluzle 11738 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ (ℤ‘1) → 1 ≤ (#‘𝐴))
4741, 46syl 17 . . . . . . . . . . . 12 (𝜑 → 1 ≤ (#‘𝐴))
48 elfzelz 12380 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...(#‘𝐴)) → 𝑘 ∈ ℤ)
4948ssriv 3640 . . . . . . . . . . . . . . . 16 (1...(#‘𝐴)) ⊆ ℤ
50 zssre 11422 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℝ
5149, 50sstri 3645 . . . . . . . . . . . . . . 15 (1...(#‘𝐴)) ⊆ ℝ
5251a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (1...(#‘𝐴)) ⊆ ℝ)
53 ressxr 10121 . . . . . . . . . . . . . 14 ℝ ⊆ ℝ*
5452, 53syl6ss 3648 . . . . . . . . . . . . 13 (𝜑 → (1...(#‘𝐴)) ⊆ ℝ*)
55 eluzelre 11736 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℝ)
5655ssriv 3640 . . . . . . . . . . . . . . 15 (ℤ𝑀) ⊆ ℝ
5734, 56syl6ss 3648 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
5857, 53syl6ss 3648 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℝ*)
59 eluzfz2 12387 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
6041, 59syl 17 . . . . . . . . . . . . 13 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
61 leisorel 13282 . . . . . . . . . . . . 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 1375 . . . . . . . . . . . 12 (𝜑 → (1 ≤ (#‘𝐴) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
6347, 62mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝐺‘1) ≤ (𝐺‘(#‘𝐴)))
6439, 60ffvelrnd 6400 . . . . . . . . . . . . . 14 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
6534, 64sseldd 3637 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
66 eluzelz 11735 . . . . . . . . . . . . 13 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(#‘𝐴)) ∈ ℤ)
6765, 66syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℤ)
68 elfz5 12372 . . . . . . . . . . . 12 (((𝐺‘1) ∈ (ℤ𝑀) ∧ (𝐺‘(#‘𝐴)) ∈ ℤ) → ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
6945, 67, 68syl2anc 694 . . . . . . . . . . 11 (𝜑 → ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(#‘𝐴))))
7063, 69mpbird 247 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))))
71 fveq2 6229 . . . . . . . . . . . . 13 (𝑘 = (𝐺‘1) → (𝐹𝑘) = (𝐹‘(𝐺‘1)))
7271eleq1d 2715 . . . . . . . . . . . 12 (𝑘 = (𝐺‘1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝐺‘1)) ∈ 𝑆))
7372imbi2d 329 . . . . . . . . . . 11 (𝑘 = (𝐺‘1) → ((𝜑 → (𝐹𝑘) ∈ 𝑆) ↔ (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)))
74 seqcoll.5 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
7574expcom 450 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))) → (𝜑 → (𝐹𝑘) ∈ 𝑆))
7673, 75vtoclga 3303 . . . . . . . . . 10 ((𝐺‘1) ∈ (𝑀...(𝐺‘(#‘𝐴))) → (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆))
7770, 76mpcom 38 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)
78 eluzelz 11735 . . . . . . . . . . . . . . . . . 18 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
7945, 78syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺‘1) ∈ ℤ)
80 peano2zm 11458 . . . . . . . . . . . . . . . . 17 ((𝐺‘1) ∈ ℤ → ((𝐺‘1) − 1) ∈ ℤ)
8179, 80syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐺‘1) − 1) ∈ ℤ)
8281zred 11520 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ∈ ℝ)
8379zred 11520 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘1) ∈ ℝ)
8467zred 11520 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
8583lem1d 10995 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘1))
8682, 83, 84, 85, 63letrd 10232 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴)))
87 eluz 11739 . . . . . . . . . . . . . . 15 ((((𝐺‘1) − 1) ∈ ℤ ∧ (𝐺‘(#‘𝐴)) ∈ ℤ) → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴))))
8881, 67, 87syl2anc 694 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(#‘𝐴))))
8986, 88mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)))
90 fzss2 12419 . . . . . . . . . . . . 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 3636 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
93 eluzel2 11730 . . . . . . . . . . . . . . 15 ((𝐺‘1) ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
9445, 93syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
95 elfzm11 12449 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝐺‘1) ∈ ℤ) → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
9694, 79, 95syl2anc 694 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
97 simp3 1083 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → 𝑘 < (𝐺‘1))
98 f1ocnv 6187 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
9937, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
100 f1of 6175 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺:𝐴1-1-onto→(1...(#‘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
10199, 100syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐺:𝐴⟶(1...(#‘𝐴)))
102101ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
103 elfznn 12408 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℕ)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ ℕ)
105104nnge1d 11101 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → 1 ≤ (𝐺𝑘))
10635adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
10754adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (1...(#‘𝐴)) ⊆ ℝ*)
10858adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐴 ⊆ ℝ*)
10943adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 1 ∈ (1...(#‘𝐴)))
110 leisorel 13282 . . . . . . . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → (1 ≤ (𝐺𝑘) ↔ (𝐺‘1) ≤ (𝐺‘(𝐺𝑘))))
112105, 111mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ (𝐺‘(𝐺𝑘)))
113 f1ocnvfv2 6573 . . . . . . . . . . . . . . . . . . 19 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
11437, 113sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
115112, 114breqtrd 4711 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ 𝑘)
11683adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘1) ∈ ℝ)
11757sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑘 ∈ ℝ)
118116, 117lenltd 10221 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → ((𝐺‘1) ≤ 𝑘 ↔ ¬ 𝑘 < (𝐺‘1)))
119115, 118mpbid 222 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ¬ 𝑘 < (𝐺‘1))
120119ex 449 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘𝐴 → ¬ 𝑘 < (𝐺‘1)))
121120con2d 129 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 < (𝐺‘1) → ¬ 𝑘𝐴))
12297, 121syl5 34 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → ¬ 𝑘𝐴))
12396, 122sylbid 230 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) → ¬ 𝑘𝐴))
124123imp 444 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → ¬ 𝑘𝐴)
12592, 124eldifd 3618 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴))
126 seqcoll.6 . . . . . . . . . 10 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
127125, 126syldan 486 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → (𝐹𝑘) = 𝑍)
12832, 33, 45, 77, 127seqid 12886 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1))) = seq(𝐺‘1)( + , 𝐹))
129128fveq1d 6231 . . . . . . 7 (𝜑 → ((seq𝑀( + , 𝐹) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)))
130 uzid 11740 . . . . . . . . 9 ((𝐺‘1) ∈ ℤ → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
13179, 130syl 17 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
132 fvres 6245 . . . . . . . 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 12854 . . . . . . . . 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 6229 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐻𝑛) = (𝐻‘1))
137 fveq2 6229 . . . . . . . . . . . . 13 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
138137fveq2d 6233 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘1)))
139136, 138eqeq12d 2666 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘1) = (𝐹‘(𝐺‘1))))
140139imbi2d 329 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))))
141 seqcoll.7 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
142141expcom 450 . . . . . . . . . 10 (𝑛 ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))))
143140, 142vtoclga 3303 . . . . . . . . 9 (1 ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1))))
14443, 143mpcom 38 . . . . . . . 8 (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))
145135, 144eqtr4d 2688 . . . . . . 7 (𝜑 → (seq(𝐺‘1)( + , 𝐹)‘(𝐺‘1)) = (𝐻‘1))
146129, 133, 1453eqtr3d 2693 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘1)) = (𝐻‘1))
147 1z 11445 . . . . . . 7 1 ∈ ℤ
148 seq1 12854 . . . . . . 7 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
149147, 148ax-mp 5 . . . . . 6 (seq1( + , 𝐻)‘1) = (𝐻‘1)
150146, 149syl6eqr 2703 . . . . 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 807 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℕ)
153 nnuz 11761 . . . . . . . . . . 11 ℕ = (ℤ‘1)
154152, 153syl6eleq 2740 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ (ℤ‘1))
155 nnz 11437 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
156155ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℤ)
157 elfzuz3 12377 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (#‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
158157adantl 481 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
159 peano2uzr 11781 . . . . . . . . . . 11 ((𝑚 ∈ ℤ ∧ (#‘𝐴) ∈ (ℤ‘(𝑚 + 1))) → (#‘𝐴) ∈ (ℤ𝑚))
160156, 158, 159syl2anc 694 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (ℤ𝑚))
161 elfzuzb 12374 . . . . . . . . . 10 (𝑚 ∈ (1...(#‘𝐴)) ↔ (𝑚 ∈ (ℤ‘1) ∧ (#‘𝐴) ∈ (ℤ𝑚)))
162154, 160, 161sylanbrc 699 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ (1...(#‘𝐴)))
163162ex 449 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → 𝑚 ∈ (1...(#‘𝐴))))
164163imim1d 82 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚)) → ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚))))
165 oveq1 6697 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq1( + , 𝐻)‘𝑚) + (𝐻‘(𝑚 + 1))))
166 simpll 805 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝜑)
167 seqcoll.1b . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
168166, 167sylan 487 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
16934ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐴 ⊆ (ℤ𝑀))
17039ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐺:(1...(#‘𝐴))⟶𝐴)
171170, 162ffvelrnd 6400 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ 𝐴)
172169, 171sseldd 3637 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ (ℤ𝑀))
173 nnre 11065 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
174173ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 ∈ ℝ)
175174ltp1d 10992 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑚 < (𝑚 + 1))
17635ad2antrr 762 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
177 simpr 476 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 + 1) ∈ (1...(#‘𝐴)))
178 isorel 6616 . . . . . . . . . . . . . . . . . 18 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(#‘𝐴)) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴)))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
179176, 162, 177, 178syl12anc 1364 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
180175, 179mpbid 222 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) < (𝐺‘(𝑚 + 1)))
181 eluzelz 11735 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑚) ∈ (ℤ𝑀) → (𝐺𝑚) ∈ ℤ)
182172, 181syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ∈ ℤ)
183170, 177ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ 𝐴)
184169, 183sseldd 3637 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ𝑀))
185 eluzelz 11735 . . . . . . . . . . . . . . . . . 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 11468 . . . . . . . . . . . . . . . . 17 (((𝐺𝑚) ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
188182, 186, 187syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
189180, 188mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1))
190 peano2zm 11458 . . . . . . . . . . . . . . . . 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 11739 . . . . . . . . . . . . . . . 16 (((𝐺𝑚) ∈ ℤ ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
193182, 191, 192syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
194189, 193mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)))
195191zred 11520 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℝ)
196186zred 11520 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℝ)
19784ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ ℝ)
198196lem1d 10995 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(𝑚 + 1)))
199 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (𝑚 + 1) ≤ (#‘𝐴))
200199adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑚 + 1) ≤ (#‘𝐴))
20154ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (1...(#‘𝐴)) ⊆ ℝ*)
20258ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝐴 ⊆ ℝ*)
20360ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (#‘𝐴) ∈ (1...(#‘𝐴)))
204 leisorel 13282 . . . . . . . . . . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝑚 + 1) ≤ (#‘𝐴) ↔ (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(#‘𝐴))))
206200, 205mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(#‘𝐴)))
207195, 196, 197, 198, 206letrd 10232 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(#‘𝐴)))
20867ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ ℤ)
209 eluz 11739 . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ↔ ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(#‘𝐴))))
211207, 210mpbird 247 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)))
212 uztrn 11742 . . . . . . . . . . . . . . . . . . 19 (((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘(𝐺𝑚)))
213211, 194, 212syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(#‘𝐴)) ∈ (ℤ‘(𝐺𝑚)))
214 fzss2 12419 . . . . . . . . . . . . . . . . . 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 3636 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺𝑚))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
217166, 74sylan 487 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
218216, 217syldan 486 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (𝑀...(𝐺𝑚))) → (𝐹𝑘) ∈ 𝑆)
219 seqcoll.c . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
220166, 219sylan 487 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
221172, 218, 220seqcl 12861 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) ∈ 𝑆)
222 simplll 813 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝜑)
223 elfzuz 12376 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)))
224 peano2uz 11779 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑚) ∈ (ℤ𝑀) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
225172, 224syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
226 uztrn 11742 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)) ∧ ((𝐺𝑚) + 1) ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
227223, 225, 226syl2anr 494 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (ℤ𝑀))
228 elfzuz3 12377 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘))
229 uztrn 11742 . . . . . . . . . . . . . . . . . 18 (((𝐺‘(#‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘)) → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘))
230211, 228, 229syl2an 493 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘))
231 elfzuzb 12374 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))) ↔ (𝑘 ∈ (ℤ𝑀) ∧ (𝐺‘(#‘𝐴)) ∈ (ℤ𝑘)))
232227, 230, 231sylanbrc 699 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴))))
233 elfzle1 12382 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺𝑚) + 1) ≤ 𝑘)
234 elfzle2 12383 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))
235233, 234jca 553 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
236155ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ ℤ)
237101ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
238 simprr 811 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘𝐴)
239237, 238ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
240 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
242 btwnnz 11491 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ 𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ)
2432423expib 1287 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℤ → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ))
244243con2d 129 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℤ → ((𝐺𝑘) ∈ ℤ → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1))))
245236, 241, 244sylc 65 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)))
24635ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
247162adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ (1...(#‘𝐴)))
248 isorel 6616 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
249246, 247, 239, 248syl12anc 1364 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
25037ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
251250, 238, 113syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
252251breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < (𝐺‘(𝐺𝑘)) ↔ (𝐺𝑚) < 𝑘))
253182adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑚) ∈ ℤ)
25434ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝐴 ⊆ (ℤ𝑀))
255254, 238sseldd 3637 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ (ℤ𝑀))
256 eluzelz 11735 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
257255, 256syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ ℤ)
258 zltp1le 11465 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺𝑚) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
259253, 257, 258syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
260249, 252, 2593bitrd 294 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
261177adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 + 1) ∈ (1...(#‘𝐴)))
262 isorel 6616 . . . . . . . . . . . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ (𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1))))
264251breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1)) ↔ 𝑘 < (𝐺‘(𝑚 + 1))))
265186adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
266 zltlem1 11468 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
267257, 265, 266syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
268263, 264, 2673bitrd 294 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
269260, 268anbi12d 747 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) ↔ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
270245, 269mtbid 313 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(#‘𝐴)) ∧ 𝑘𝐴)) → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
271270expr 642 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑘𝐴 → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
272271con2d 129 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)) → ¬ 𝑘𝐴))
273235, 272syl5 34 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ¬ 𝑘𝐴))
274273imp 444 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → ¬ 𝑘𝐴)
275232, 274eldifd 3618 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴))
276222, 275, 126syl2anc 694 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐹𝑘) = 𝑍)
277168, 172, 194, 221, 276seqid2 12887 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq𝑀( + , 𝐹)‘((𝐺‘(𝑚 + 1)) − 1)))
278277oveq1d 6705 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐻𝑛) = (𝐻‘(𝑚 + 1)))
280 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
281280fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘(𝑚 + 1))))
282279, 281eqeq12d 2666 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑚 + 1) → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
283282imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))))
284283, 142vtoclga 3303 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ (1...(#‘𝐴)) → (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
285284impcom 445 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
286285adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
287286oveq2d 6706 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
28894ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → 𝑀 ∈ ℤ)
289186zcnd 11521 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℂ)
290 ax-1cn 10032 . . . . . . . . . . . . . . 15 1 ∈ ℂ
291 npcan 10328 . . . . . . . . . . . . . . 15 (((𝐺‘(𝑚 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
292289, 290, 291sylancl 695 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
293 uztrn 11742 . . . . . . . . . . . . . . . 16 ((((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ∧ (𝐺𝑚) ∈ (ℤ𝑀)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
294194, 172, 293syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
295 eluzp1p1 11751 . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ‘(𝑀 + 1)))
298 seqm1 12858 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 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 2696 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))))
301 seqp1 12856 . . . . . . . . . . . 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 2666 . . . . . . . . . 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 236 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(#‘𝐴))) → ((seq𝑀( + , 𝐹)‘(𝐺𝑚)) = (seq1( + , 𝐻)‘𝑚) → (seq𝑀( + , 𝐹)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻)‘(𝑚 + 1))))
305304ex 449 . . . . . . . 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 47 . . . . . 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 450 . . . . 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 11076 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...(#‘𝐴)) → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))))
3113, 310mpcom 38 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cdif 3604  wss 3607   class class class wbr 4685  ccnv 5142  cres 5145  wf 5922  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  cc 9972  cr 9973  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  cz 11415  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842
This theorem is referenced by:  seqcoll2  13287  summolem2a  14490  prodmolem2a  14708
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