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

Theorem offsplitfpar 7960
Description: Express the function operation map f by the functions defined in fsplit 7957 and fpar 7956. (Contributed by AV, 4-Jan-2024.)
Hypotheses
Ref Expression
fsplitfpar.h 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
fsplitfpar.s 𝑆 = ((1st ↾ I ) ↾ 𝐴)
Assertion
Ref Expression
offsplitfpar (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = (𝐹f + 𝐺))

Proof of Theorem offsplitfpar
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fsplitfpar.h . . . . 5 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
2 fsplitfpar.s . . . . 5 𝑆 = ((1st ↾ I ) ↾ 𝐴)
31, 2fsplitfpar 7959 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))
43coeq2d 5771 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ( + ∘ (𝐻𝑆)) = ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)))
543ad2ant1 1132 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)))
6 dffn3 6613 . . . . . . 7 ( + Fn 𝐶+ :𝐶⟶ran + )
76biimpi 215 . . . . . 6 ( + Fn 𝐶+ :𝐶⟶ran + )
87adantr 481 . . . . 5 (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
983ad2ant3 1134 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
10 simpl3r 1228 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (ran 𝐹 × ran 𝐺) ⊆ 𝐶)
11 simp1l 1196 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
12 fnfvelrn 6958 . . . . . . 7 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
1311, 12sylan 580 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
14 simp1r 1197 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
15 fnfvelrn 6958 . . . . . . 7 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1614, 15sylan 580 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1713, 16opelxpd 5627 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (ran 𝐹 × ran 𝐺))
1810, 17sseldd 3922 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ 𝐶)
199, 18cofmpt 7004 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)))
20 df-ov 7278 . . . . 5 ((𝐹𝑎) + (𝐺𝑎)) = ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)
2120eqcomi 2747 . . . 4 ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩) = ((𝐹𝑎) + (𝐺𝑎))
2221mpteq2i 5179 . . 3 (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2319, 22eqtrdi 2794 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
24 offval3 7825 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))))
25 fndm 6536 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
26 fndm 6536 . . . . . . . 8 (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
2725, 26ineqan12d 4148 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐴))
28 inidm 4152 . . . . . . 7 (𝐴𝐴) = 𝐴
2927, 28eqtrdi 2794 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
3029mpteq1d 5169 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3124, 30sylan9eqr 2800 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊)) → (𝐹f + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3231eqcomd 2744 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊)) → (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝐹f + 𝐺))
33323adant3 1131 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝐹f + 𝐺))
345, 23, 333eqtrd 2782 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = (𝐹f + 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  wss 3887  cop 4567  cmpt 5157   I cid 5488   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  ccom 5593   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  1st c1st 7829  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-1st 7831  df-2nd 7832
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator