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

Theorem seqhomo 12788
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 12291 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2686 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6148 . . . . . . . 8 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
65fveq2d 6152 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
7 fveq2 6148 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑀))
86, 7eqeq12d 2636 . . . . . 6 (𝑥 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
94, 8imbi12d 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 6148 . . . . . . . 8 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1312fveq2d 6152 . . . . . . 7 (𝑥 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
14 fveq2 6148 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑛))
1513, 14eqeq12d 2636 . . . . . 6 (𝑥 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
1611, 15imbi12d 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 6148 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
2019fveq2d 6152 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
21 fveq2 6148 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
2220, 21eqeq12d 2636 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
2318, 22imbi12d 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 6148 . . . . . . . 8 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2726fveq2d 6152 . . . . . . 7 (𝑥 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
28 fveq2 6148 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑁))
2927, 28eqeq12d 2636 . . . . . 6 (𝑥 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
3025, 29imbi12d 334 . . . . 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 330 . . . 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 12290 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
34 seqhomo.5 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
3534ralrimiva 2960 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
36 fveq2 6148 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
3736fveq2d 6152 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
38 fveq2 6148 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
3937, 38eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4039rspcv 3291 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4133, 35, 40sylc 65 . . . . . . 7 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
42 eluzel2 11636 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
43 seq1 12754 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4544fveq2d 6152 . . . . . . 7 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹𝑀)))
46 seq1 12754 . . . . . . . 8 (𝑀 ∈ ℤ → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
471, 42, 463syl 18 . . . . . . 7 (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
4841, 45, 473eqtr4d 2665 . . . . . 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 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
52 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
53 peano2fzr 12296 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5451, 52, 53syl2anc 692 . . . . . . . . 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 6611 . . . . . . . . . 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 12756 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5958ad2antrl 763 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6059fveq2d 6152 . . . . . . . . . . . 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 2965 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
6362adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
64 elfzuz3 12281 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝑛))
65 fzss2 12323 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6654, 64, 653syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6766sselda 3583 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
68 seqhomo.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
6968adantlr 750 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
7067, 69syldan 487 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹𝑥) ∈ 𝑆)
71 seqhomo.1 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7271adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7351, 70, 72seqcl 12761 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
7468ralrimiva 2960 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
7574adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
76 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
7776eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
7877rspcv 3291 . . . . . . . . . . . . . . 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 6611 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
8180fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
82 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
8382oveq1d 6619 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)))
8481, 83eqeq12d 2636 . . . . . . . . . . . . . . 15 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦))))
85 oveq2 6612 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
8685fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
87 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
8887oveq2d 6620 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
8986, 88eqeq12d 2636 . . . . . . . . . . . . . . 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 3306 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
9476fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
95 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
9694, 95eqeq12d 2636 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
9796rspcv 3291 . . . . . . . . . . . . . 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 6620 . . . . . . . . . . . 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 2659 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
101 seqp1 12756 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
102101ad2antrl 763 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
103100, 102eqeq12d 2636 . . . . . . . . . 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 451 . . . . 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 11690 . . 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 384   = wceq 1480  wcel 1987  wral 2907  wss 3555  cfv 5847  (class class class)co 6604  1c1 9881   + caddc 9883  cz 11321  cuz 11631  ...cfz 12268  seqcseq 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-seq 12742
This theorem is referenced by:  seqfeq4  12790  seqdistr  12792  seqof  12798  fsumrelem  14466  efcj  14747  gsumwmhm  17303  gsumzmhm  18258  elqaalem2  23979  logfac  24251  gamcvg2lem  24685  prmorcht  24804  pclogsum  24840
  Copyright terms: Public domain W3C validator