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Theorem seqhomo 12888
 Description: Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqhomo.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqhomo.2 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
seqhomo.3 (𝜑𝑁 ∈ (ℤ𝑀))
seqhomo.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
seqhomo.5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
Assertion
Ref Expression
seqhomo (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐻,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺   𝑥, + ,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦
Allowed substitution hint:   𝐺(𝑦)

Proof of Theorem seqhomo
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqhomo.3 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12387 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2718 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
65fveq2d 6233 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
7 fveq2 6229 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑀))
86, 7eqeq12d 2666 . . . . . 6 (𝑥 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
94, 8imbi12d 333 . . . . 5 (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
109imbi2d 329 . . . 4 (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))))
11 eleq1 2718 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1312fveq2d 6233 . . . . . . 7 (𝑥 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
14 fveq2 6229 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑛))
1513, 14eqeq12d 2666 . . . . . 6 (𝑥 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
1611, 15imbi12d 333 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
1716imbi2d 329 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))))
18 eleq1 2718 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 6229 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
2019fveq2d 6233 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
21 fveq2 6229 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
2220, 21eqeq12d 2666 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
2318, 22imbi12d 333 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
2423imbi2d 329 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))))
25 eleq1 2718 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2726fveq2d 6233 . . . . . . 7 (𝑥 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
28 fveq2 6229 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑁))
2927, 28eqeq12d 2666 . . . . . 6 (𝑥 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
3025, 29imbi12d 333 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
3130imbi2d 329 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))))
32 eluzfz1 12386 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
34 seqhomo.5 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
3534ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
36 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
3736fveq2d 6233 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
38 fveq2 6229 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
3937, 38eqeq12d 2666 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4039rspcv 3336 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4133, 35, 40sylc 65 . . . . . . 7 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
42 eluzel2 11730 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
43 seq1 12854 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4544fveq2d 6233 . . . . . . 7 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹𝑀)))
46 seq1 12854 . . . . . . . 8 (𝑀 ∈ ℤ → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
471, 42, 463syl 18 . . . . . . 7 (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
4841, 45, 473eqtr4d 2695 . . . . . 6 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))
4948a1d 25 . . . . 5 (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
5049a1i 11 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
51 simprl 809 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
52 simprr 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
53 peano2fzr 12392 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5451, 52, 53syl2anc 694 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
5554expr 642 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
5655imim1d 82 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
57 oveq1 6697 . . . . . . . . . 10 ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
58 seqp1 12856 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5958ad2antrl 764 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6059fveq2d 6233 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
61 seqhomo.4 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
6261ralrimivva 3000 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
6362adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
64 elfzuz3 12377 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝑛))
65 fzss2 12419 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6654, 64, 653syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6766sselda 3636 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
68 seqhomo.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
6968adantlr 751 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
7067, 69syldan 486 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹𝑥) ∈ 𝑆)
71 seqhomo.1 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7271adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7351, 70, 72seqcl 12861 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
7468ralrimiva 2995 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
7574adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
76 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
7776eleq1d 2715 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
7877rspcv 3336 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆))
7952, 75, 78sylc 65 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
80 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
8180fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
82 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
8382oveq1d 6705 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)))
8481, 83eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦))))
85 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
8685fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
87 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
8887oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
8986, 88eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
9084, 89rspc2v 3353 . . . . . . . . . . . . . 14 (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
9173, 79, 90syl2anc 694 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
9263, 91mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
9335adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
9476fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
95 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
9694, 95eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
9796rspcv 3336 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
9852, 93, 97sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
9998oveq2d 6706 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
10060, 92, 993eqtrd 2689 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
101 seqp1 12856 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
102101ad2antrl 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
103100, 102eqeq12d 2666 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) ↔ ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1)))))
10457, 103syl5ibr 236 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
105104expr 642 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
106105a2d 29 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
10756, 106syld 47 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
108107expcom 450 . . . . 5 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))))
109108a2d 29 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))))
11010, 17, 24, 31, 50, 109uzind4 11784 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
1111, 110mpcom 38 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
1123, 111mpd 15 1 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  ‘cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  ℤcz 11415  ℤ≥cuz 11725  ...cfz 12364  seqcseq 12841 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-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:  seqfeq4  12890  seqdistr  12892  seqof  12898  fsumrelem  14583  efcj  14866  gsumwmhm  17429  gsumzmhm  18383  elqaalem2  24120  logfac  24392  gamcvg2lem  24830  prmorcht  24949  pclogsum  24985
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