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Theorem offsplitfpar 8143
Description: Express the function operation map f by the functions defined in fsplit 8141 and fpar 8140. (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 8142 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))
43coeq2d 5876 . . 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 6749 . . . . . . 7 ( + Fn 𝐶+ :𝐶⟶ran + )
76biimpi 216 . . . . . 6 ( + Fn 𝐶+ :𝐶⟶ran + )
87adantr 480 . . . . 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 7100 . . . . . . 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 7100 . . . . . . 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 5728 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (ran 𝐹 × ran 𝐺))
1810, 17sseldd 3996 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ 𝐶)
199, 18cofmpt 7152 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)))
20 df-ov 7434 . . . . 5 ((𝐹𝑎) + (𝐺𝑎)) = ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)
2120eqcomi 2744 . . . 4 ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩) = ((𝐹𝑎) + (𝐺𝑎))
2221mpteq2i 5253 . . 3 (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2319, 22eqtrdi 2791 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
24 offval3 8006 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))))
25 fndm 6672 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
26 fndm 6672 . . . . . . . 8 (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
2725, 26ineqan12d 4230 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐴))
28 inidm 4235 . . . . . . 7 (𝐴𝐴) = 𝐴
2927, 28eqtrdi 2791 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
3029mpteq1d 5243 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3124, 30sylan9eqr 2797 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊)) → (𝐹f + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3231eqcomd 2741 . . 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 2779 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 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  cop 4637  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8013  df-2nd 8014
This theorem is referenced by: (None)
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