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Theorem seqhomo 13040
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 12540 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2825 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6350 . . . . . . . 8 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
65fveq2d 6354 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
7 fveq2 6350 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑀))
86, 7eqeq12d 2773 . . . . . 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 2825 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 6350 . . . . . . . 8 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1312fveq2d 6354 . . . . . . 7 (𝑥 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
14 fveq2 6350 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑛))
1513, 14eqeq12d 2773 . . . . . 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 2825 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 6350 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
2019fveq2d 6354 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
21 fveq2 6350 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
2220, 21eqeq12d 2773 . . . . . 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 2825 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 6350 . . . . . . . 8 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2726fveq2d 6354 . . . . . . 7 (𝑥 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
28 fveq2 6350 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑁))
2927, 28eqeq12d 2773 . . . . . 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 12539 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
34 seqhomo.5 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
3534ralrimiva 3102 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
36 fveq2 6350 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
3736fveq2d 6354 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
38 fveq2 6350 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
3937, 38eqeq12d 2773 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4039rspcv 3443 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
4133, 35, 40sylc 65 . . . . . . 7 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
42 eluzel2 11882 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
43 seq1 13006 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4544fveq2d 6354 . . . . . . 7 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹𝑀)))
46 seq1 13006 . . . . . . . 8 (𝑀 ∈ ℤ → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
471, 42, 463syl 18 . . . . . . 7 (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
4841, 45, 473eqtr4d 2802 . . . . . 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 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
52 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
53 peano2fzr 12545 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5451, 52, 53syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
5554expr 644 . . . . . . . 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 6818 . . . . . . . . . 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 13008 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5958ad2antrl 766 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6059fveq2d 6354 . . . . . . . . . . . 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 3107 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
6362adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
64 elfzuz3 12530 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝑛))
65 fzss2 12572 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6654, 64, 653syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6766sselda 3742 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
68 seqhomo.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
6968adantlr 753 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
7067, 69syldan 488 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹𝑥) ∈ 𝑆)
71 seqhomo.1 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7271adantlr 753 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7351, 70, 72seqcl 13013 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
7468ralrimiva 3102 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
7574adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) ∈ 𝑆)
76 fveq2 6350 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
7776eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
7877rspcv 3443 . . . . . . . . . . . . . . 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 6818 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
8180fveq2d 6354 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
82 fveq2 6350 . . . . . . . . . . . . . . . . 17 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
8382oveq1d 6826 . . . . . . . . . . . . . . . 16 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)))
8481, 83eqeq12d 2773 . . . . . . . . . . . . . . 15 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦))))
85 oveq2 6819 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
8685fveq2d 6354 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
87 fveq2 6350 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
8887oveq2d 6827 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
8986, 88eqeq12d 2773 . . . . . . . . . . . . . . 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 3459 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
9476fveq2d 6354 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
95 fveq2 6350 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
9694, 95eqeq12d 2773 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
9796rspcv 3443 . . . . . . . . . . . . . 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 6827 . . . . . . . . . . . 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 2796 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
101 seqp1 13008 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
102101ad2antrl 766 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
103100, 102eqeq12d 2773 . . . . . . . . . 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 644 . . . . . . . 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 11937 . . 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 1630  wcel 2137  wral 3048  wss 3713  cfv 6047  (class class class)co 6811  1c1 10127   + caddc 10129  cz 11567  cuz 11877  ...cfz 12517  seqcseq 12993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-seq 12994
This theorem is referenced by:  seqfeq4  13042  seqdistr  13044  seqof  13050  fsumrelem  14736  efcj  15019  gsumwmhm  17581  gsumzmhm  18535  elqaalem2  24272  logfac  24544  gamcvg2lem  24982  prmorcht  25101  pclogsum  25137
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