| Step | Hyp | Ref
| Expression |
| 1 | | seqf1o.4 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 2 | | eluzfz2 13572 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
| 3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
| 4 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀)) |
| 5 | 4 | oveq2d 7447 |
. . . . 5
⊢ (𝑚 = 𝑀 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀))) |
| 6 | 4 | oveq1d 7446 |
. . . . 5
⊢ (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))) |
| 7 | 5, 6 | eqeq12d 2753 |
. . . 4
⊢ (𝑚 = 𝑀 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))) |
| 8 | 7 | imbi2d 340 |
. . 3
⊢ (𝑚 = 𝑀 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))))) |
| 9 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛)) |
| 10 | 9 | oveq2d 7447 |
. . . . 5
⊢ (𝑚 = 𝑛 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛))) |
| 11 | 9 | oveq1d 7446 |
. . . . 5
⊢ (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾))) |
| 12 | 10, 11 | eqeq12d 2753 |
. . . 4
⊢ (𝑚 = 𝑛 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)))) |
| 13 | 12 | imbi2d 340 |
. . 3
⊢ (𝑚 = 𝑛 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾))))) |
| 14 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1))) |
| 15 | 14 | oveq2d 7447 |
. . . . 5
⊢ (𝑚 = (𝑛 + 1) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1)))) |
| 16 | 14 | oveq1d 7446 |
. . . . 5
⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))) |
| 17 | 15, 16 | eqeq12d 2753 |
. . . 4
⊢ (𝑚 = (𝑛 + 1) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))) |
| 18 | 17 | imbi2d 340 |
. . 3
⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))))) |
| 19 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁)) |
| 20 | 19 | oveq2d 7447 |
. . . . 5
⊢ (𝑚 = 𝑁 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁))) |
| 21 | 19 | oveq1d 7446 |
. . . . 5
⊢ (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))) |
| 22 | 20, 21 | eqeq12d 2753 |
. . . 4
⊢ (𝑚 = 𝑁 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))) |
| 23 | 22 | imbi2d 340 |
. . 3
⊢ (𝑚 = 𝑁 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))))) |
| 24 | | seqf1o.2 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
| 25 | | seqf1olem2a.1 |
. . . . . 6
⊢ (𝜑 → 𝐺:𝐴⟶𝐶) |
| 26 | | seqf1olem2a.3 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ 𝐴) |
| 27 | 25, 26 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝐶) |
| 28 | | eluzel2 12883 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 29 | | seq1 14055 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀)) |
| 30 | 1, 28, 29 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀)) |
| 31 | | seqf1olem2a.4 |
. . . . . . . 8
⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴) |
| 32 | | eluzfz1 13571 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
| 33 | 1, 32 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
| 34 | 31, 33 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ 𝐴) |
| 35 | 25, 34 | ffvelcdmd 7105 |
. . . . . 6
⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝐶) |
| 36 | 30, 35 | eqeltrd 2841 |
. . . . 5
⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶) |
| 37 | 24, 27, 36 | caovcomd 7629 |
. . . 4
⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))) |
| 38 | 37 | a1i 11 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))) |
| 39 | | oveq1 7438 |
. . . . . 6
⊢ (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1)))) |
| 40 | | elfzouz 13703 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 41 | 40 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 42 | | seqp1 14057 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))) |
| 43 | 41, 42 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))) |
| 44 | 43 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))) |
| 45 | | seqf1o.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
| 46 | 45 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
| 47 | | seqf1o.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ⊆ 𝑆) |
| 48 | 47, 27 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝑆) |
| 49 | 48 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝑆) |
| 50 | 47 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐶 ⊆ 𝑆) |
| 51 | 50 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶 ⊆ 𝑆) |
| 52 | 25 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴⟶𝐶) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴⟶𝐶) |
| 54 | | elfzouz2 13714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛)) |
| 55 | 54 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛)) |
| 56 | | fzss2 13604 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
| 58 | 31 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴) |
| 59 | 57, 58 | sstrd 3994 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴) |
| 60 | 59 | sselda 3983 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ 𝐴) |
| 61 | 53, 60 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝐶) |
| 62 | 51, 61 | sseldd 3984 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝑆) |
| 63 | | seqf1o.1 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
| 64 | 63 | adantlr 715 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
| 65 | 41, 62, 64 | seqcl 14063 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆) |
| 66 | | fzofzp1 13803 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
| 67 | 66 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
| 68 | 58, 67 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴) |
| 69 | 52, 68 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶) |
| 70 | 50, 69 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆) |
| 71 | 46, 49, 65, 70 | caovassd 7632 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))) |
| 72 | 44, 71 | eqtr4d 2780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1)))) |
| 73 | 46, 65, 70, 49 | caovassd 7632 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)))) |
| 74 | 43 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾))) |
| 75 | 46, 65, 49, 70 | caovassd 7632 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1))))) |
| 76 | 24 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
| 77 | 27 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝐶) |
| 78 | 76, 69, 77 | caovcomd 7629 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)) = ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))) |
| 79 | 78 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1))))) |
| 80 | 75, 79 | eqtr4d 2780 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)))) |
| 81 | 73, 74, 80 | 3eqtr4d 2787 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1)))) |
| 82 | 72, 81 | eqeq12d 2753 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) ↔ (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))))) |
| 83 | 39, 82 | imbitrrid 246 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))) |
| 84 | 83 | expcom 413 |
. . . 4
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))))) |
| 85 | 84 | a2d 29 |
. . 3
⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾))) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))))) |
| 86 | 8, 13, 18, 23, 38, 85 | fzind2 13824 |
. 2
⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))) |
| 87 | 3, 86 | mpcom 38 |
1
⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))) |