Step |
Hyp |
Ref |
Expression |
1 |
|
ofrn.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
|
ofrn.2 |
⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 ) |
3 |
|
ofrn.3 |
⊢ ( 𝜑 → + : ( 𝐵 × 𝐵 ) ⟶ 𝐶 ) |
4 |
|
ofrn.4 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
5 |
1
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
6 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) → 𝑎 ∈ 𝐴 ) |
7 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑎 ) ∈ ran 𝐹 ) |
8 |
5 6 7
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) → ( 𝐹 ‘ 𝑎 ) ∈ ran 𝐹 ) |
9 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐴 ) |
10 |
|
fnfvelrn |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑎 ) ∈ ran 𝐺 ) |
11 |
9 6 10
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) → ( 𝐺 ‘ 𝑎 ) ∈ ran 𝐺 ) |
12 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) → 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) |
13 |
|
rspceov |
⊢ ( ( ( 𝐹 ‘ 𝑎 ) ∈ ran 𝐹 ∧ ( 𝐺 ‘ 𝑎 ) ∈ ran 𝐺 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) → ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) ) |
14 |
8 11 12 13
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) → ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) ) |
15 |
14
|
rexlimdvaa |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) → ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) ) ) |
16 |
15
|
ss2abdv |
⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) } ⊆ { 𝑧 ∣ ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) } ) |
17 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
18 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑎 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑎 ) = ( 𝐺 ‘ 𝑎 ) ) |
20 |
5 9 4 4 17 18 19
|
offval |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) |
21 |
20
|
rneqd |
⊢ ( 𝜑 → ran ( 𝐹 ∘f + 𝐺 ) = ran ( 𝑎 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) ) |
22 |
|
eqid |
⊢ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) |
23 |
22
|
rnmpt |
⊢ ran ( 𝑎 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) ) = { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) } |
24 |
21 23
|
eqtrdi |
⊢ ( 𝜑 → ran ( 𝐹 ∘f + 𝐺 ) = { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( ( 𝐹 ‘ 𝑎 ) + ( 𝐺 ‘ 𝑎 ) ) } ) |
25 |
3
|
ffnd |
⊢ ( 𝜑 → + Fn ( 𝐵 × 𝐵 ) ) |
26 |
1
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐵 ) |
27 |
2
|
frnd |
⊢ ( 𝜑 → ran 𝐺 ⊆ 𝐵 ) |
28 |
|
xpss12 |
⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐵 ) → ( ran 𝐹 × ran 𝐺 ) ⊆ ( 𝐵 × 𝐵 ) ) |
29 |
26 27 28
|
syl2anc |
⊢ ( 𝜑 → ( ran 𝐹 × ran 𝐺 ) ⊆ ( 𝐵 × 𝐵 ) ) |
30 |
|
ovelimab |
⊢ ( ( + Fn ( 𝐵 × 𝐵 ) ∧ ( ran 𝐹 × ran 𝐺 ) ⊆ ( 𝐵 × 𝐵 ) ) → ( 𝑧 ∈ ( + “ ( ran 𝐹 × ran 𝐺 ) ) ↔ ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) ) ) |
31 |
25 29 30
|
syl2anc |
⊢ ( 𝜑 → ( 𝑧 ∈ ( + “ ( ran 𝐹 × ran 𝐺 ) ) ↔ ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) ) ) |
32 |
31
|
abbi2dv |
⊢ ( 𝜑 → ( + “ ( ran 𝐹 × ran 𝐺 ) ) = { 𝑧 ∣ ∃ 𝑥 ∈ ran 𝐹 ∃ 𝑦 ∈ ran 𝐺 𝑧 = ( 𝑥 + 𝑦 ) } ) |
33 |
16 24 32
|
3sstr4d |
⊢ ( 𝜑 → ran ( 𝐹 ∘f + 𝐺 ) ⊆ ( + “ ( ran 𝐹 × ran 𝐺 ) ) ) |