| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ofccat.1 |
⊢ ( 𝜑 → 𝐸 ∈ Word 𝑆 ) |
| 2 |
|
ofccat.2 |
⊢ ( 𝜑 → 𝐹 ∈ Word 𝑆 ) |
| 3 |
|
ofccat.3 |
⊢ ( 𝜑 → 𝐺 ∈ Word 𝑇 ) |
| 4 |
|
ofccat.4 |
⊢ ( 𝜑 → 𝐻 ∈ Word 𝑇 ) |
| 5 |
|
ofccat.5 |
⊢ ( 𝜑 → ( ♯ ‘ 𝐸 ) = ( ♯ ‘ 𝐺 ) ) |
| 6 |
|
ofccat.6 |
⊢ ( 𝜑 → ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐻 ) ) |
| 7 |
|
wrdf |
⊢ ( 𝐸 ∈ Word 𝑆 → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑆 ) |
| 8 |
|
ffn |
⊢ ( 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑆 → 𝐸 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 9 |
1 7 8
|
3syl |
⊢ ( 𝜑 → 𝐸 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 10 |
|
wrdf |
⊢ ( 𝐺 ∈ Word 𝑇 → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ⟶ 𝑇 ) |
| 11 |
|
ffn |
⊢ ( 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ⟶ 𝑇 → 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) |
| 12 |
3 10 11
|
3syl |
⊢ ( 𝜑 → 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) |
| 13 |
5
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐸 ) ) = ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) |
| 14 |
13
|
fneq2d |
⊢ ( 𝜑 → ( 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ↔ 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) ) |
| 15 |
12 14
|
mpbird |
⊢ ( 𝜑 → 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 16 |
|
ovexd |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ∈ V ) |
| 17 |
|
inidm |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ∩ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) |
| 18 |
9 15 16 16 17
|
offn |
⊢ ( 𝜑 → ( 𝐸 ∘f 𝑅 𝐺 ) Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 19 |
|
hashfn |
⊢ ( ( 𝐸 ∘f 𝑅 𝐺 ) Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) → ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ) |
| 20 |
18 19
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ) |
| 21 |
|
wrdfin |
⊢ ( 𝐸 ∈ Word 𝑆 → 𝐸 ∈ Fin ) |
| 22 |
|
hashcl |
⊢ ( 𝐸 ∈ Fin → ( ♯ ‘ 𝐸 ) ∈ ℕ0 ) |
| 23 |
1 21 22
|
3syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐸 ) ∈ ℕ0 ) |
| 24 |
|
hashfzo0 |
⊢ ( ( ♯ ‘ 𝐸 ) ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) = ( ♯ ‘ 𝐸 ) ) |
| 25 |
23 24
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) = ( ♯ ‘ 𝐸 ) ) |
| 26 |
20 25
|
eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) = ( ♯ ‘ 𝐸 ) ) |
| 27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) = ( ♯ ‘ 𝐸 ) ) |
| 28 |
27
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 29 |
28
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ↔ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ) |
| 30 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝐸 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 31 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 32 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ∈ V ) |
| 33 |
29
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 34 |
|
fnfvof |
⊢ ( ( ( 𝐸 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ∧ 𝐺 Fn ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ∧ ( ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ∈ V ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ) → ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) = ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) ) |
| 35 |
30 31 32 33 34
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) = ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) ) |
| 36 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) = ( ♯ ‘ 𝐸 ) ) |
| 37 |
36
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) = ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) |
| 38 |
37
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) = ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) |
| 39 |
|
wrdf |
⊢ ( 𝐹 ∈ Word 𝑆 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ 𝑆 ) |
| 40 |
|
ffn |
⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ 𝑆 → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 41 |
2 39 40
|
3syl |
⊢ ( 𝜑 → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 42 |
41
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 43 |
|
wrdf |
⊢ ( 𝐻 ∈ Word 𝑇 → 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ⟶ 𝑇 ) |
| 44 |
|
ffn |
⊢ ( 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ⟶ 𝑇 → 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ) |
| 45 |
4 43 44
|
3syl |
⊢ ( 𝜑 → 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ) |
| 46 |
6
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ) |
| 47 |
46
|
fneq2d |
⊢ ( 𝜑 → ( 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐻 ) ) ) ) |
| 48 |
45 47
|
mpbird |
⊢ ( 𝜑 → 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 49 |
48
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 50 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∈ V ) |
| 51 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) |
| 52 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) |
| 53 |
28
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 54 |
52 53
|
neleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
| 55 |
23
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ♯ ‘ 𝐸 ) ∈ ℕ0 ) |
| 56 |
55
|
nn0zd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ♯ ‘ 𝐸 ) ∈ ℤ ) |
| 57 |
|
wrdfin |
⊢ ( 𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin ) |
| 58 |
|
hashcl |
⊢ ( 𝐹 ∈ Fin → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
| 59 |
2 57 58
|
3syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
| 60 |
59
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
| 61 |
60
|
nn0zd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ♯ ‘ 𝐹 ) ∈ ℤ ) |
| 62 |
|
fzocatel |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) ∧ ( ( ♯ ‘ 𝐸 ) ∈ ℤ ∧ ( ♯ ‘ 𝐹 ) ∈ ℤ ) ) → ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 63 |
51 54 56 61 62
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 64 |
|
fnfvof |
⊢ ( ( ( 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐻 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∈ V ∧ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) = ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) |
| 65 |
42 49 50 63 64
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) = ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) |
| 66 |
38 65
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) ∧ ¬ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) → ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) |
| 67 |
29 35 66
|
ifbieq12d2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) = if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) , ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) |
| 68 |
67
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) , ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) ) |
| 69 |
|
ovex |
⊢ ( 𝐸 ∘f 𝑅 𝐺 ) ∈ V |
| 70 |
|
ovex |
⊢ ( 𝐹 ∘f 𝑅 𝐻 ) ∈ V |
| 71 |
|
ccatfval |
⊢ ( ( ( 𝐸 ∘f 𝑅 𝐺 ) ∈ V ∧ ( 𝐹 ∘f 𝑅 𝐻 ) ∈ V ) → ( ( 𝐸 ∘f 𝑅 𝐺 ) ++ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) + ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) ) |
| 72 |
69 70 71
|
mp2an |
⊢ ( ( 𝐸 ∘f 𝑅 𝐺 ) ++ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) + ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) |
| 73 |
|
ovexd |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∈ V ) |
| 74 |
|
inidm |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∩ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐹 ) ) |
| 75 |
41 48 73 73 74
|
offn |
⊢ ( 𝜑 → ( 𝐹 ∘f 𝑅 𝐻 ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 76 |
|
hashfn |
⊢ ( ( 𝐹 ∘f 𝑅 𝐻 ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
| 77 |
75 76
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
| 78 |
|
hashfzo0 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) = ( ♯ ‘ 𝐹 ) ) |
| 79 |
59 78
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) = ( ♯ ‘ 𝐹 ) ) |
| 80 |
77 79
|
eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( ♯ ‘ 𝐹 ) ) |
| 81 |
26 80
|
oveq12d |
⊢ ( 𝜑 → ( ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) + ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) = ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) |
| 82 |
81
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) + ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) |
| 83 |
82
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) + ( ♯ ‘ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) ) |
| 84 |
72 83
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝐸 ∘f 𝑅 𝐺 ) ++ ( 𝐹 ∘f 𝑅 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) , ( ( 𝐸 ∘f 𝑅 𝐺 ) ‘ 𝑖 ) , ( ( 𝐹 ∘f 𝑅 𝐻 ) ‘ ( 𝑖 − ( ♯ ‘ ( 𝐸 ∘f 𝑅 𝐺 ) ) ) ) ) ) ) |
| 85 |
|
ovexd |
⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ∈ V ) |
| 86 |
|
fvex |
⊢ ( 𝐸 ‘ 𝑖 ) ∈ V |
| 87 |
|
fvex |
⊢ ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ∈ V |
| 88 |
86 87
|
ifex |
⊢ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ∈ V |
| 89 |
88
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ∈ V ) |
| 90 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑖 ) ∈ V |
| 91 |
|
fvex |
⊢ ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ∈ V |
| 92 |
90 91
|
ifex |
⊢ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ∈ V |
| 93 |
92
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ∈ V ) |
| 94 |
|
ccatfval |
⊢ ( ( 𝐸 ∈ Word 𝑆 ∧ 𝐹 ∈ Word 𝑆 ) → ( 𝐸 ++ 𝐹 ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) |
| 95 |
1 2 94
|
syl2anc |
⊢ ( 𝜑 → ( 𝐸 ++ 𝐹 ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) |
| 96 |
|
ccatfval |
⊢ ( ( 𝐺 ∈ Word 𝑇 ∧ 𝐻 ∈ Word 𝑇 ) → ( 𝐺 ++ 𝐻 ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐺 ) + ( ♯ ‘ 𝐻 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) |
| 97 |
3 4 96
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ++ 𝐻 ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐺 ) + ( ♯ ‘ 𝐻 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) |
| 98 |
5 6
|
oveq12d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) = ( ( ♯ ‘ 𝐺 ) + ( ♯ ‘ 𝐻 ) ) ) |
| 99 |
98
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐺 ) + ( ♯ ‘ 𝐻 ) ) ) ) |
| 100 |
99
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐺 ) + ( ♯ ‘ 𝐻 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) |
| 101 |
97 100
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐺 ++ 𝐻 ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) |
| 102 |
85 89 93 95 101
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐸 ++ 𝐹 ) ∘f 𝑅 ( 𝐺 ++ 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) ) |
| 103 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( ♯ ‘ 𝐸 ) = ( ♯ ‘ 𝐺 ) ) |
| 104 |
103
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 0 ..^ ( ♯ ‘ 𝐸 ) ) = ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) |
| 105 |
104
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ↔ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) ) ) |
| 106 |
103
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑖 − ( ♯ ‘ 𝐸 ) ) = ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) |
| 107 |
106
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) = ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) |
| 108 |
105 107
|
ifbieq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) = if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) |
| 109 |
108
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ) → ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) = ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) |
| 110 |
109
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐺 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐺 ) ) ) ) ) ) ) |
| 111 |
102 110
|
eqtr4d |
⊢ ( 𝜑 → ( ( 𝐸 ++ 𝐹 ) ∘f 𝑅 ( 𝐺 ++ 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) ) |
| 112 |
|
ovif12 |
⊢ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) = if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) , ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) |
| 113 |
112
|
mpteq2i |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ ( if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐸 ‘ 𝑖 ) , ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) 𝑅 if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( 𝐺 ‘ 𝑖 ) , ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) , ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) |
| 114 |
111 113
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝐸 ++ 𝐹 ) ∘f 𝑅 ( 𝐺 ++ 𝐻 ) ) = ( 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐸 ) + ( ♯ ‘ 𝐹 ) ) ) ↦ if ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) , ( ( 𝐸 ‘ 𝑖 ) 𝑅 ( 𝐺 ‘ 𝑖 ) ) , ( ( 𝐹 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) 𝑅 ( 𝐻 ‘ ( 𝑖 − ( ♯ ‘ 𝐸 ) ) ) ) ) ) ) |
| 115 |
68 84 114
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( 𝐸 ++ 𝐹 ) ∘f 𝑅 ( 𝐺 ++ 𝐻 ) ) = ( ( 𝐸 ∘f 𝑅 𝐺 ) ++ ( 𝐹 ∘f 𝑅 𝐻 ) ) ) |