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Theorem iseqcoll 10212
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 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
seqcoll.hcl ((𝜑𝑘 ∈ (ℤ‘1)) → (𝐻𝑘) ∈ 𝑆)
seqcoll.6 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
seqcoll.7 ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
Assertion
Ref Expression
iseqcoll (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑘,𝐻,𝑛   𝑘,𝑀,𝑛   + ,𝑘,𝑛   𝜑,𝑘,𝑛   𝑆,𝑘,𝑛   𝑘,𝑍,𝑛
Allowed substitution hints:   𝑁(𝑘,𝑛)

Proof of Theorem iseqcoll
Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqcoll.3 . 2 (𝜑𝑁 ∈ (1...(♯‘𝐴)))
2 elfznn 9437 . . . 4 (𝑁 ∈ (1...(♯‘𝐴)) → 𝑁 ∈ ℕ)
31, 2syl 14 . . 3 (𝜑𝑁 ∈ ℕ)
4 eleq1 2150 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...(♯‘𝐴)) ↔ 1 ∈ (1...(♯‘𝐴))))
5 2fveq3 5294 . . . . . . 7 (𝑦 = 1 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)))
6 fveq2 5289 . . . . . . 7 (𝑦 = 1 → (seq1( + , 𝐻, 𝑆)‘𝑦) = (seq1( + , 𝐻, 𝑆)‘1))
75, 6eqeq12d 2102 . . . . . 6 (𝑦 = 1 → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (seq1( + , 𝐻, 𝑆)‘1)))
84, 7imbi12d 232 . . . . 5 (𝑦 = 1 → ((𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦)) ↔ (1 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (seq1( + , 𝐻, 𝑆)‘1))))
98imbi2d 228 . . . 4 (𝑦 = 1 → ((𝜑 → (𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦))) ↔ (𝜑 → (1 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (seq1( + , 𝐻, 𝑆)‘1)))))
10 eleq1 2150 . . . . . 6 (𝑦 = 𝑚 → (𝑦 ∈ (1...(♯‘𝐴)) ↔ 𝑚 ∈ (1...(♯‘𝐴))))
11 2fveq3 5294 . . . . . . 7 (𝑦 = 𝑚 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)))
12 fveq2 5289 . . . . . . 7 (𝑦 = 𝑚 → (seq1( + , 𝐻, 𝑆)‘𝑦) = (seq1( + , 𝐻, 𝑆)‘𝑚))
1311, 12eqeq12d 2102 . . . . . 6 (𝑦 = 𝑚 → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)))
1410, 13imbi12d 232 . . . . 5 (𝑦 = 𝑚 → ((𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦)) ↔ (𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚))))
1514imbi2d 228 . . . 4 (𝑦 = 𝑚 → ((𝜑 → (𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦))) ↔ (𝜑 → (𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)))))
16 eleq1 2150 . . . . . 6 (𝑦 = (𝑚 + 1) → (𝑦 ∈ (1...(♯‘𝐴)) ↔ (𝑚 + 1) ∈ (1...(♯‘𝐴))))
17 2fveq3 5294 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))))
18 fveq2 5289 . . . . . . 7 (𝑦 = (𝑚 + 1) → (seq1( + , 𝐻, 𝑆)‘𝑦) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)))
1917, 18eqeq12d 2102 . . . . . 6 (𝑦 = (𝑚 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1))))
2016, 19imbi12d 232 . . . . 5 (𝑦 = (𝑚 + 1) → ((𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦)) ↔ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)))))
2120imbi2d 228 . . . 4 (𝑦 = (𝑚 + 1) → ((𝜑 → (𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦))) ↔ (𝜑 → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1))))))
22 eleq1 2150 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...(♯‘𝐴)) ↔ 𝑁 ∈ (1...(♯‘𝐴))))
23 2fveq3 5294 . . . . . . 7 (𝑦 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)))
24 fveq2 5289 . . . . . . 7 (𝑦 = 𝑁 → (seq1( + , 𝐻, 𝑆)‘𝑦) = (seq1( + , 𝐻, 𝑆)‘𝑁))
2523, 24eqeq12d 2102 . . . . . 6 (𝑦 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁)))
2622, 25imbi12d 232 . . . . 5 (𝑦 = 𝑁 → ((𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦)) ↔ (𝑁 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁))))
2726imbi2d 228 . . . 4 (𝑦 = 𝑁 → ((𝜑 → (𝑦 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑦)) = (seq1( + , 𝐻, 𝑆)‘𝑦))) ↔ (𝜑 → (𝑁 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁)))))
28 seqcoll.1 . . . . . . . . 9 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
29 seqcoll.a . . . . . . . . 9 (𝜑𝑍𝑆)
30 seqcoll.4 . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℤ𝑀))
31 seqcoll.2 . . . . . . . . . . . . 13 (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
32 isof1o 5568 . . . . . . . . . . . . 13 (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
3331, 32syl 14 . . . . . . . . . . . 12 (𝜑𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
34 f1of 5237 . . . . . . . . . . . 12 (𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝐺:(1...(♯‘𝐴))⟶𝐴)
3533, 34syl 14 . . . . . . . . . . 11 (𝜑𝐺:(1...(♯‘𝐴))⟶𝐴)
36 elfzuz2 9412 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(♯‘𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
371, 36syl 14 . . . . . . . . . . . 12 (𝜑 → (♯‘𝐴) ∈ (ℤ‘1))
38 eluzfz1 9414 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘1) → 1 ∈ (1...(♯‘𝐴)))
3937, 38syl 14 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...(♯‘𝐴)))
4035, 39ffvelrnd 5419 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ 𝐴)
4130, 40sseldd 3024 . . . . . . . . 9 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
42 eluzle 9000 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ (ℤ‘1) → 1 ≤ (♯‘𝐴))
4337, 42syl 14 . . . . . . . . . . . 12 (𝜑 → 1 ≤ (♯‘𝐴))
44 elfzelz 9409 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...(♯‘𝐴)) → 𝑘 ∈ ℤ)
4544ssriv 3027 . . . . . . . . . . . . . . . 16 (1...(♯‘𝐴)) ⊆ ℤ
46 zssre 8727 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℝ
4745, 46sstri 3032 . . . . . . . . . . . . . . 15 (1...(♯‘𝐴)) ⊆ ℝ
4847a1i 9 . . . . . . . . . . . . . 14 (𝜑 → (1...(♯‘𝐴)) ⊆ ℝ)
49 ressxr 7510 . . . . . . . . . . . . . 14 ℝ ⊆ ℝ*
5048, 49syl6ss 3035 . . . . . . . . . . . . 13 (𝜑 → (1...(♯‘𝐴)) ⊆ ℝ*)
51 eluzelre 8998 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℝ)
5251ssriv 3027 . . . . . . . . . . . . . . 15 (ℤ𝑀) ⊆ ℝ
5330, 52syl6ss 3035 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
5453, 49syl6ss 3035 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℝ*)
55 eluzfz2 9415 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ (ℤ‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
5637, 55syl 14 . . . . . . . . . . . . 13 (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
57 leisorel 10207 . . . . . . . . . . . . 13 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((1...(♯‘𝐴)) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ (1 ∈ (1...(♯‘𝐴)) ∧ (♯‘𝐴) ∈ (1...(♯‘𝐴)))) → (1 ≤ (♯‘𝐴) ↔ (𝐺‘1) ≤ (𝐺‘(♯‘𝐴))))
5831, 50, 54, 39, 56, 57syl122anc 1183 . . . . . . . . . . . 12 (𝜑 → (1 ≤ (♯‘𝐴) ↔ (𝐺‘1) ≤ (𝐺‘(♯‘𝐴))))
5943, 58mpbid 145 . . . . . . . . . . 11 (𝜑 → (𝐺‘1) ≤ (𝐺‘(♯‘𝐴)))
6035, 56ffvelrnd 5419 . . . . . . . . . . . . . 14 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
6130, 60sseldd 3024 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀))
62 eluzelz 8997 . . . . . . . . . . . . 13 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℤ)
6361, 62syl 14 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℤ)
64 elfz5 9401 . . . . . . . . . . . 12 (((𝐺‘1) ∈ (ℤ𝑀) ∧ (𝐺‘(♯‘𝐴)) ∈ ℤ) → ((𝐺‘1) ∈ (𝑀...(𝐺‘(♯‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(♯‘𝐴))))
6541, 63, 64syl2anc 403 . . . . . . . . . . 11 (𝜑 → ((𝐺‘1) ∈ (𝑀...(𝐺‘(♯‘𝐴))) ↔ (𝐺‘1) ≤ (𝐺‘(♯‘𝐴))))
6659, 65mpbird 165 . . . . . . . . . 10 (𝜑 → (𝐺‘1) ∈ (𝑀...(𝐺‘(♯‘𝐴))))
67 fveq2 5289 . . . . . . . . . . . . 13 (𝑘 = (𝐺‘1) → (𝐹𝑘) = (𝐹‘(𝐺‘1)))
6867eleq1d 2156 . . . . . . . . . . . 12 (𝑘 = (𝐺‘1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝐺‘1)) ∈ 𝑆))
6968imbi2d 228 . . . . . . . . . . 11 (𝑘 = (𝐺‘1) → ((𝜑 → (𝐹𝑘) ∈ 𝑆) ↔ (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)))
70 elfzuz 9405 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴))) → 𝑘 ∈ (ℤ𝑀))
71 seqcoll.5 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
7271expcom 114 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (𝐹𝑘) ∈ 𝑆))
7370, 72syl 14 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴))) → (𝜑 → (𝐹𝑘) ∈ 𝑆))
7469, 73vtoclga 2685 . . . . . . . . . 10 ((𝐺‘1) ∈ (𝑀...(𝐺‘(♯‘𝐴))) → (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆))
7566, 74mpcom 36 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺‘1)) ∈ 𝑆)
76 eluzelz 8997 . . . . . . . . . . . . . . . . . 18 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
7741, 76syl 14 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺‘1) ∈ ℤ)
78 peano2zm 8758 . . . . . . . . . . . . . . . . 17 ((𝐺‘1) ∈ ℤ → ((𝐺‘1) − 1) ∈ ℤ)
7977, 78syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐺‘1) − 1) ∈ ℤ)
8079zred 8838 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ∈ ℝ)
8177zred 8838 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘1) ∈ ℝ)
8263zred 8838 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
8381lem1d 8366 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘1))
8480, 81, 82, 83, 59letrd 7586 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘1) − 1) ≤ (𝐺‘(♯‘𝐴)))
85 eluz 9001 . . . . . . . . . . . . . . 15 ((((𝐺‘1) − 1) ∈ ℤ ∧ (𝐺‘(♯‘𝐴)) ∈ ℤ) → ((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(♯‘𝐴))))
8679, 63, 85syl2anc 403 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) ↔ ((𝐺‘1) − 1) ≤ (𝐺‘(♯‘𝐴))))
8784, 86mpbird 165 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)))
88 fzss2 9446 . . . . . . . . . . . . 13 ((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘1) − 1)) → (𝑀...((𝐺‘1) − 1)) ⊆ (𝑀...(𝐺‘(♯‘𝐴))))
8987, 88syl 14 . . . . . . . . . . . 12 (𝜑 → (𝑀...((𝐺‘1) − 1)) ⊆ (𝑀...(𝐺‘(♯‘𝐴))))
9089sselda 3023 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴))))
91 eluzel2 8993 . . . . . . . . . . . . . . 15 ((𝐺‘1) ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
9241, 91syl 14 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
93 elfzm11 9472 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝐺‘1) ∈ ℤ) → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
9492, 77, 93syl2anc 403 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) ↔ (𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1))))
95 simp3 945 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → 𝑘 < (𝐺‘1))
9681adantr 270 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (𝐺‘1) ∈ ℝ)
9753sselda 3023 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘 ∈ ℝ)
98 f1ocnv 5250 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝐺:𝐴1-1-onto→(1...(♯‘𝐴)))
9933, 98syl 14 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺:𝐴1-1-onto→(1...(♯‘𝐴)))
100 f1of 5237 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺:𝐴1-1-onto→(1...(♯‘𝐴)) → 𝐺:𝐴⟶(1...(♯‘𝐴)))
10199, 100syl 14 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐺:𝐴⟶(1...(♯‘𝐴)))
102101ffvelrnda 5418 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ (1...(♯‘𝐴)))
103 elfznn 9437 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ∈ ℕ)
104102, 103syl 14 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (𝐺𝑘) ∈ ℕ)
105104nnge1d 8436 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → 1 ≤ (𝐺𝑘))
10631adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
10750adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → (1...(♯‘𝐴)) ⊆ ℝ*)
10854adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 𝐴 ⊆ ℝ*)
10939adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → 1 ∈ (1...(♯‘𝐴)))
110 leisorel 10207 . . . . . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → (1 ≤ (𝐺𝑘) ↔ (𝐺‘1) ≤ (𝐺‘(𝐺𝑘))))
112105, 111mpbid 145 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ (𝐺‘(𝐺𝑘)))
113 f1ocnvfv2 5539 . . . . . . . . . . . . . . . . . . 19 ((𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
11433, 113sylan 277 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
115112, 114breqtrd 3861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (𝐺‘1) ≤ 𝑘)
11696, 97, 115lensymd 7584 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ¬ 𝑘 < (𝐺‘1))
117116ex 113 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘𝐴 → ¬ 𝑘 < (𝐺‘1)))
118117con2d 589 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 < (𝐺‘1) → ¬ 𝑘𝐴))
11995, 118syl5 32 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ ℤ ∧ 𝑀𝑘𝑘 < (𝐺‘1)) → ¬ 𝑘𝐴))
12094, 119sylbid 148 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝑀...((𝐺‘1) − 1)) → ¬ 𝑘𝐴))
121120imp 122 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → ¬ 𝑘𝐴)
12290, 121eldifd 3007 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴))
123 seqcoll.6 . . . . . . . . . 10 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
124122, 123syldan 276 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...((𝐺‘1) − 1))) → (𝐹𝑘) = 𝑍)
125 seqcoll.c . . . . . . . . 9 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
12628, 29, 41, 75, 124, 71, 125iseqid 9904 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ‘(𝐺‘1))) = seq(𝐺‘1)( + , 𝐹, 𝑆))
127126fveq1d 5291 . . . . . . 7 (𝜑 → ((seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq(𝐺‘1)( + , 𝐹, 𝑆)‘(𝐺‘1)))
128 uzid 9002 . . . . . . . . 9 ((𝐺‘1) ∈ ℤ → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
12977, 128syl 14 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ (ℤ‘(𝐺‘1)))
130 fvres 5313 . . . . . . . 8 ((𝐺‘1) ∈ (ℤ‘(𝐺‘1)) → ((seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)))
131129, 130syl 14 . . . . . . 7 (𝜑 → ((seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ‘(𝐺‘1)))‘(𝐺‘1)) = (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)))
13292adantr 270 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑀 ∈ ℤ)
133 eluzelz 8997 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ‘(𝐺‘1)) → 𝑘 ∈ ℤ)
134133adantl 271 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑘 ∈ ℤ)
135132zred 8838 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑀 ∈ ℝ)
13681adantr 270 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ ℝ)
137134zred 8838 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑘 ∈ ℝ)
138 eluzle 9000 . . . . . . . . . . . . . 14 ((𝐺‘1) ∈ (ℤ𝑀) → 𝑀 ≤ (𝐺‘1))
13941, 138syl 14 . . . . . . . . . . . . 13 (𝜑𝑀 ≤ (𝐺‘1))
140139adantr 270 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑀 ≤ (𝐺‘1))
141 eluzle 9000 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ‘(𝐺‘1)) → (𝐺‘1) ≤ 𝑘)
142141adantl 271 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ≤ 𝑘)
143135, 136, 137, 140, 142letrd 7586 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑀𝑘)
144 eluz2 8994 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀𝑘))
145132, 134, 143, 144syl3anbrc 1127 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → 𝑘 ∈ (ℤ𝑀))
146145, 71syldan 276 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ‘(𝐺‘1))) → (𝐹𝑘) ∈ 𝑆)
14777, 146, 125iseq1 9840 . . . . . . . 8 (𝜑 → (seq(𝐺‘1)( + , 𝐹, 𝑆)‘(𝐺‘1)) = (𝐹‘(𝐺‘1)))
148 fveq2 5289 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐻𝑛) = (𝐻‘1))
149 2fveq3 5294 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘1)))
150148, 149eqeq12d 2102 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘1) = (𝐹‘(𝐺‘1))))
151150imbi2d 228 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))))
152 seqcoll.7 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
153152expcom 114 . . . . . . . . . 10 (𝑛 ∈ (1...(♯‘𝐴)) → (𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))))
154151, 153vtoclga 2685 . . . . . . . . 9 (1 ∈ (1...(♯‘𝐴)) → (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1))))
15539, 154mpcom 36 . . . . . . . 8 (𝜑 → (𝐻‘1) = (𝐹‘(𝐺‘1)))
156147, 155eqtr4d 2123 . . . . . . 7 (𝜑 → (seq(𝐺‘1)( + , 𝐹, 𝑆)‘(𝐺‘1)) = (𝐻‘1))
157127, 131, 1563eqtr3d 2128 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (𝐻‘1))
158 1zzd 8747 . . . . . . 7 (𝜑 → 1 ∈ ℤ)
159 seqcoll.hcl . . . . . . 7 ((𝜑𝑘 ∈ (ℤ‘1)) → (𝐻𝑘) ∈ 𝑆)
160158, 159, 125iseq1 9840 . . . . . 6 (𝜑 → (seq1( + , 𝐻, 𝑆)‘1) = (𝐻‘1))
161157, 160eqtr4d 2123 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (seq1( + , 𝐻, 𝑆)‘1))
162161a1d 22 . . . 4 (𝜑 → (1 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘1)) = (seq1( + , 𝐻, 𝑆)‘1)))
163 simplr 497 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 ∈ ℕ)
164 nnuz 9023 . . . . . . . . . . 11 ℕ = (ℤ‘1)
165163, 164syl6eleq 2180 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 ∈ (ℤ‘1))
166 nnz 8739 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
167166ad2antlr 473 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 ∈ ℤ)
168 elfzuz3 9406 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (♯‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
169168adantl 271 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (♯‘𝐴) ∈ (ℤ‘(𝑚 + 1)))
170 peano2uzr 9042 . . . . . . . . . . 11 ((𝑚 ∈ ℤ ∧ (♯‘𝐴) ∈ (ℤ‘(𝑚 + 1))) → (♯‘𝐴) ∈ (ℤ𝑚))
171167, 169, 170syl2anc 403 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (♯‘𝐴) ∈ (ℤ𝑚))
172 elfzuzb 9403 . . . . . . . . . 10 (𝑚 ∈ (1...(♯‘𝐴)) ↔ (𝑚 ∈ (ℤ‘1) ∧ (♯‘𝐴) ∈ (ℤ𝑚)))
173165, 171, 172sylanbrc 408 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 ∈ (1...(♯‘𝐴)))
174173ex 113 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → 𝑚 ∈ (1...(♯‘𝐴))))
175174imim1d 74 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚))))
176 oveq1 5641 . . . . . . . . . 10 ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq1( + , 𝐻, 𝑆)‘𝑚) + (𝐻‘(𝑚 + 1))))
177 simpll 496 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝜑)
178 seqcoll.1b . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
179177, 178sylan 277 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
18030ad2antrr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ (ℤ𝑀))
18135ad2antrr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝐺:(1...(♯‘𝐴))⟶𝐴)
182181, 173ffvelrnd 5419 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺𝑚) ∈ 𝐴)
183180, 182sseldd 3024 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺𝑚) ∈ (ℤ𝑀))
184 nnre 8401 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
185184ad2antlr 473 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 ∈ ℝ)
186185ltp1d 8363 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑚 < (𝑚 + 1))
18731ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
188 simpr 108 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝑚 + 1) ∈ (1...(♯‘𝐴)))
189 isorel 5569 . . . . . . . . . . . . . . . . . 18 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(♯‘𝐴)) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴)))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
190187, 173, 188, 189syl12anc 1172 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝑚 < (𝑚 + 1) ↔ (𝐺𝑚) < (𝐺‘(𝑚 + 1))))
191186, 190mpbid 145 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺𝑚) < (𝐺‘(𝑚 + 1)))
192 eluzelz 8997 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑚) ∈ (ℤ𝑀) → (𝐺𝑚) ∈ ℤ)
193183, 192syl 14 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺𝑚) ∈ ℤ)
194181, 188ffvelrnd 5419 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ 𝐴)
195180, 194sseldd 3024 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ𝑀))
196 eluzelz 8997 . . . . . . . . . . . . . . . . . 18 ((𝐺‘(𝑚 + 1)) ∈ (ℤ𝑀) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
197195, 196syl 14 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
198 zltlem1 8777 . . . . . . . . . . . . . . . . 17 (((𝐺𝑚) ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
199193, 197, 198syl2anc 403 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺𝑚) < (𝐺‘(𝑚 + 1)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
200191, 199mpbid 145 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1))
201 peano2zm 8758 . . . . . . . . . . . . . . . . 17 ((𝐺‘(𝑚 + 1)) ∈ ℤ → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ)
202197, 201syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ)
203 eluz 9001 . . . . . . . . . . . . . . . 16 (((𝐺𝑚) ∈ ℤ ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
204193, 202, 203syl2anc 403 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ↔ (𝐺𝑚) ≤ ((𝐺‘(𝑚 + 1)) − 1)))
205200, 204mpbird 165 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)))
206177, 71sylan 277 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
207177, 125sylan 277 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
208183, 206, 207iseqcl 9846 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) ∈ 𝑆)
209 simplll 500 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝜑)
210 elfzuz 9405 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)))
211 peano2uz 9040 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑚) ∈ (ℤ𝑀) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
212183, 211syl 14 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺𝑚) + 1) ∈ (ℤ𝑀))
213 uztrn 9004 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (ℤ‘((𝐺𝑚) + 1)) ∧ ((𝐺𝑚) + 1) ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
214210, 212, 213syl2anr 284 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (ℤ𝑀))
215202zred 8838 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ ℝ)
216197zred 8838 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℝ)
21782ad2antrr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
218216lem1d 8366 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(𝑚 + 1)))
219 elfzle2 9411 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (𝑚 + 1) ≤ (♯‘𝐴))
220219adantl 271 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝑚 + 1) ≤ (♯‘𝐴))
22150ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (1...(♯‘𝐴)) ⊆ ℝ*)
22254ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ ℝ*)
22356ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
224 leisorel 10207 . . . . . . . . . . . . . . . . . . . . . 22 ((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))))
225187, 221, 222, 188, 223, 224syl122anc 1183 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝑚 + 1) ≤ (♯‘𝐴) ↔ (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(♯‘𝐴))))
226220, 225mpbid 145 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ≤ (𝐺‘(♯‘𝐴)))
227215, 216, 217, 218, 226letrd 7586 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(♯‘𝐴)))
22863ad2antrr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(♯‘𝐴)) ∈ ℤ)
229 eluz 9001 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺‘(𝑚 + 1)) − 1) ∈ ℤ ∧ (𝐺‘(♯‘𝐴)) ∈ ℤ) → ((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ↔ ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(♯‘𝐴))))
230202, 228, 229syl2anc 403 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ↔ ((𝐺‘(𝑚 + 1)) − 1) ≤ (𝐺‘(♯‘𝐴))))
231227, 230mpbird 165 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)))
232 elfzuz3 9406 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘))
233 uztrn 9004 . . . . . . . . . . . . . . . . . 18 (((𝐺‘(♯‘𝐴)) ∈ (ℤ‘((𝐺‘(𝑚 + 1)) − 1)) ∧ ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑘)) → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑘))
234231, 232, 233syl2an 283 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑘))
235 elfzuzb 9403 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴))) ↔ (𝑘 ∈ (ℤ𝑀) ∧ (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑘)))
236214, 234, 235sylanbrc 408 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴))))
237166ad2antlr 473 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ ℤ)
238101ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:𝐴⟶(1...(♯‘𝐴)))
239 simprr 499 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝑘𝐴)
240238, 239ffvelrnd 5419 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(♯‘𝐴)))
241 elfzelz 9409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ∈ ℤ)
242240, 241syl 14 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
243 btwnnz 8810 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℤ ∧ 𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ)
2442433expib 1146 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℤ → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) → ¬ (𝐺𝑘) ∈ ℤ))
245244con2d 589 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℤ → ((𝐺𝑘) ∈ ℤ → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1))))
246237, 242, 245sylc 61 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ¬ (𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)))
24731ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
248173adantrr 463 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝑚 ∈ (1...(♯‘𝐴)))
249 isorel 5569 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ (𝑚 ∈ (1...(♯‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(♯‘𝐴)))) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
250247, 248, 240, 249syl12anc 1172 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ (𝐺𝑚) < (𝐺‘(𝐺𝑘))))
25133ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
252251, 239, 113syl2anc 403 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
253252breq2d 3849 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < (𝐺‘(𝐺𝑘)) ↔ (𝐺𝑚) < 𝑘))
254193adantrr 463 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝐺𝑚) ∈ ℤ)
25530ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝐴 ⊆ (ℤ𝑀))
256255, 239sseldd 3024 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ (ℤ𝑀))
257 eluzelz 8997 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
258256, 257syl 14 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → 𝑘 ∈ ℤ)
259 zltp1le 8774 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺𝑚) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
260254, 258, 259syl2anc 403 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑚) < 𝑘 ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
261250, 253, 2603bitrd 212 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 < (𝐺𝑘) ↔ ((𝐺𝑚) + 1) ≤ 𝑘))
262188adantrr 463 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝑚 + 1) ∈ (1...(♯‘𝐴)))
263 isorel 5569 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((𝐺𝑘) ∈ (1...(♯‘𝐴)) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴)))) → ((𝐺𝑘) < (𝑚 + 1) ↔ (𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1))))
264247, 240, 262, 263syl12anc 1172 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ (𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1))))
265252breq1d 3847 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺‘(𝐺𝑘)) < (𝐺‘(𝑚 + 1)) ↔ 𝑘 < (𝐺‘(𝑚 + 1))))
266197adantrr 463 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝐺‘(𝑚 + 1)) ∈ ℤ)
267 zltlem1 8777 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℤ ∧ (𝐺‘(𝑚 + 1)) ∈ ℤ) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
268258, 266, 267syl2anc 403 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → (𝑘 < (𝐺‘(𝑚 + 1)) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
269264, 265, 2683bitrd 212 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝐺𝑘) < (𝑚 + 1) ↔ 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
270261, 269anbi12d 457 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ((𝑚 < (𝐺𝑘) ∧ (𝐺𝑘) < (𝑚 + 1)) ↔ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
271246, 270mtbid 632 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑚 + 1) ∈ (1...(♯‘𝐴)) ∧ 𝑘𝐴)) → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
272271expr 367 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝑘𝐴 → ¬ (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))))
273272con2d 589 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)) → ¬ 𝑘𝐴))
274 elfzle1 9410 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → ((𝐺𝑚) + 1) ≤ 𝑘)
275 elfzle2 9411 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → 𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1))
276274, 275jca 300 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1)) → (((𝐺𝑚) + 1) ≤ 𝑘𝑘 ≤ ((𝐺‘(𝑚 + 1)) − 1)))
277273, 276impel 274 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → ¬ 𝑘𝐴)
278236, 277eldifd 3007 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴))
279209, 278, 123syl2anc 403 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (((𝐺𝑚) + 1)...((𝐺‘(𝑚 + 1)) − 1))) → (𝐹𝑘) = 𝑍)
280179, 183, 205, 208, 279, 206, 207iseqid2 9906 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq𝑀( + , 𝐹, 𝑆)‘((𝐺‘(𝑚 + 1)) − 1)))
281280oveq1d 5649 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐹‘(𝐺‘(𝑚 + 1)))) = ((seq𝑀( + , 𝐹, 𝑆)‘((𝐺‘(𝑚 + 1)) − 1)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
282 fveq2 5289 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐻𝑛) = (𝐻‘(𝑚 + 1)))
283 2fveq3 5294 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑚 + 1) → (𝐹‘(𝐺𝑛)) = (𝐹‘(𝐺‘(𝑚 + 1))))
284282, 283eqeq12d 2102 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑚 + 1) → ((𝐻𝑛) = (𝐹‘(𝐺𝑛)) ↔ (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
285284imbi2d 228 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐻𝑛) = (𝐹‘(𝐺𝑛))) ↔ (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))))
286285, 153vtoclga 2685 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (𝜑 → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1)))))
287286impcom 123 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
288287adantlr 461 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐻‘(𝑚 + 1)) = (𝐹‘(𝐺‘(𝑚 + 1))))
289288oveq2d 5650 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
29092ad2antrr 472 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → 𝑀 ∈ ℤ)
291197zcnd 8839 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ ℂ)
292 ax-1cn 7417 . . . . . . . . . . . . . . 15 1 ∈ ℂ
293 npcan 7670 . . . . . . . . . . . . . . 15 (((𝐺‘(𝑚 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
294291, 292, 293sylancl 404 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) + 1) = (𝐺‘(𝑚 + 1)))
295 uztrn 9004 . . . . . . . . . . . . . . . 16 ((((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ‘(𝐺𝑚)) ∧ (𝐺𝑚) ∈ (ℤ𝑀)) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
296205, 183, 295syl2anc 403 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀))
297 eluzp1p1 9013 . . . . . . . . . . . . . . 15 (((𝐺‘(𝑚 + 1)) − 1) ∈ (ℤ𝑀) → (((𝐺‘(𝑚 + 1)) − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
298296, 297syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (((𝐺‘(𝑚 + 1)) − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
299294, 298eqeltrrd 2165 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (𝐺‘(𝑚 + 1)) ∈ (ℤ‘(𝑀 + 1)))
300290, 299, 206, 207iseqm1 9853 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘((𝐺‘(𝑚 + 1)) − 1)) + (𝐹‘(𝐺‘(𝑚 + 1)))))
301281, 289, 3003eqtr4rd 2131 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))))
302177, 159sylan 277 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) ∧ 𝑘 ∈ (ℤ‘1)) → (𝐻𝑘) ∈ 𝑆)
303165, 302, 207iseqp1 9847 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)) = ((seq1( + , 𝐻, 𝑆)‘𝑚) + (𝐻‘(𝑚 + 1))))
304301, 303eqeq12d 2102 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) + (𝐻‘(𝑚 + 1))) = ((seq1( + , 𝐻, 𝑆)‘𝑚) + (𝐻‘(𝑚 + 1)))))
305176, 304syl5ibr 154 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(♯‘𝐴))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1))))
306305ex 113 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)))))
307306a2d 26 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)))))
308175, 307syld 44 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1)))))
309308expcom 114 . . . . 5 (𝑚 ∈ ℕ → (𝜑 → ((𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚)) → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1))))))
310309a2d 26 . . . 4 (𝑚 ∈ ℕ → ((𝜑 → (𝑚 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑚)) = (seq1( + , 𝐻, 𝑆)‘𝑚))) → (𝜑 → ((𝑚 + 1) ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺‘(𝑚 + 1))) = (seq1( + , 𝐻, 𝑆)‘(𝑚 + 1))))))
3119, 15, 21, 27, 162, 310nnind 8410 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁))))
3123, 311mpcom 36 . 2 (𝜑 → (𝑁 ∈ (1...(♯‘𝐴)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁)))
3131, 312mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wcel 1438  cdif 2994  wss 2997   class class class wbr 3837  ccnv 4427  cres 4430  wf 4998  1-1-ontowf1o 5001  cfv 5002   Isom wiso 5003  (class class class)co 5634  cc 7327  cr 7328  1c1 7330   + caddc 7332  *cxr 7500   < clt 7501  cle 7502  cmin 7632  cn 8394  cz 8720  cuz 8988  ...cfz 9393  seqcseq4 9816  chash 10148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394  df-fzo 9519  df-iseq 9818
This theorem is referenced by:  isummolem2a  10735
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