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Theorem offsplitfpar 7793
 Description: Express the function operation map ∘f by the functions defined in fsplit 7790 and fpar 7789. (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 7792 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))
43coeq2d 5709 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ( + ∘ (𝐻𝑆)) = ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)))
543ad2ant1 1129 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)))
6 dffn3 6501 . . . . . . 7 ( + Fn 𝐶+ :𝐶⟶ran + )
76biimpi 218 . . . . . 6 ( + Fn 𝐶+ :𝐶⟶ran + )
87adantr 483 . . . . 5 (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
983ad2ant3 1131 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
10 simpl3r 1225 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (ran 𝐹 × ran 𝐺) ⊆ 𝐶)
11 simp1l 1193 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
12 fnfvelrn 6824 . . . . . . 7 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
1311, 12sylan 582 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
14 simp1r 1194 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
15 fnfvelrn 6824 . . . . . . 7 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1614, 15sylan 582 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1713, 16opelxpd 5569 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (ran 𝐹 × ran 𝐺))
1810, 17sseldd 3947 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ 𝐶)
199, 18cofmpt 6870 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)))
20 df-ov 7136 . . . . 5 ((𝐹𝑎) + (𝐺𝑎)) = ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)
2120eqcomi 2829 . . . 4 ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩) = ((𝐹𝑎) + (𝐺𝑎))
2221mpteq2i 5134 . . 3 (𝑎𝐴 ↦ ( + ‘⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2319, 22syl6eq 2871 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
24 offval3 7661 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))))
25 fndm 6431 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
26 fndm 6431 . . . . . . . 8 (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
2725, 26ineqan12d 4169 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐴))
28 inidm 4173 . . . . . . 7 (𝐴𝐴) = 𝐴
2927, 28syl6eq 2871 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
3029mpteq1d 5131 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3124, 30sylan9eqr 2877 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊)) → (𝐹f + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
3231eqcomd 2826 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊)) → (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝐹f + 𝐺))
33323adant3 1128 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝐹f + 𝐺))
345, 23, 333eqtrd 2859 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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3473   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4549   ↦ cmpt 5122   I cid 5435   × cxp 5529  ◡ccnv 5530  dom cdm 5531  ran crn 5532   ↾ cres 5533   ∘ ccom 5535   Fn wfn 6326  ⟶wf 6327  ‘cfv 6331  (class class class)co 7133   ∘f cof 7385  1st c1st 7665  2nd c2nd 7666 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-1st 7667  df-2nd 7668 This theorem is referenced by: (None)
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