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Theorem offsplitfpar 8105
Description: Express the function operation map ∘f by the functions defined in fsplit 8103 and fpar 8102. (Contributed by AV, 4-Jan-2024.)
Hypotheses
Ref Expression
fsplitfpar.h 𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))
fsplitfpar.s 𝑆 = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)
Assertion
Ref Expression
offsplitfpar (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Proof of Theorem offsplitfpar
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fsplitfpar.h . . . . 5 𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))
2 fsplitfpar.s . . . . 5 𝑆 = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)
31, 2fsplitfpar 8104 . . . 4 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (𝐻 ∘ 𝑆) = (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩))
43coeq2d 5863 . . 3 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
543ad2ant1 1134 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
6 dffn3 6731 . . . . . . 7 ( + Fn 𝐢 ↔ + :𝐢⟢ran + )
76biimpi 215 . . . . . 6 ( + Fn 𝐢 β†’ + :𝐢⟢ran + )
87adantr 482 . . . . 5 (( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢) β†’ + :𝐢⟢ran + )
983ad2ant3 1136 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ + :𝐢⟢ran + )
10 simpl3r 1230 . . . . 5 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)
11 simp1l 1198 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ 𝐹 Fn 𝐴)
12 fnfvelrn 7083 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ ran 𝐹)
1311, 12sylan 581 . . . . . 6 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ ran 𝐹)
14 simp1r 1199 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ 𝐺 Fn 𝐴)
15 fnfvelrn 7083 . . . . . . 7 ((𝐺 Fn 𝐴 ∧ π‘Ž ∈ 𝐴) β†’ (πΊβ€˜π‘Ž) ∈ ran 𝐺)
1614, 15sylan 581 . . . . . 6 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (πΊβ€˜π‘Ž) ∈ ran 𝐺)
1713, 16opelxpd 5716 . . . . 5 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩ ∈ (ran 𝐹 Γ— ran 𝐺))
1810, 17sseldd 3984 . . . 4 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩ ∈ 𝐢)
199, 18cofmpt 7130 . . 3 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
20 df-ov 7412 . . . . 5 ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž)) = ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)
2120eqcomi 2742 . . . 4 ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩) = ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))
2221mpteq2i 5254 . . 3 (π‘Ž ∈ 𝐴 ↦ ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž)))
2319, 22eqtrdi 2789 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
24 offval3 7969 . . . . 5 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ (𝐹 ∘f + 𝐺) = (π‘Ž ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
25 fndm 6653 . . . . . . . 8 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
26 fndm 6653 . . . . . . . 8 (𝐺 Fn 𝐴 β†’ dom 𝐺 = 𝐴)
2725, 26ineqan12d 4215 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
28 inidm 4219 . . . . . . 7 (𝐴 ∩ 𝐴) = 𝐴
2927, 28eqtrdi 2789 . . . . . 6 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (dom 𝐹 ∩ dom 𝐺) = 𝐴)
3029mpteq1d 5244 . . . . 5 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (π‘Ž ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
3124, 30sylan9eqr 2795 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š)) β†’ (𝐹 ∘f + 𝐺) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
3231eqcomd 2739 . . 3 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š)) β†’ (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (𝐹 ∘f + 𝐺))
33323adant3 1133 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (𝐹 ∘f + 𝐺))
345, 23, 333eqtrd 2777 1 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  βŸ¨cop 4635   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668  1st c1st 7973  2nd c2nd 7974
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-1st 7975  df-2nd 7976
This theorem is referenced by: (None)
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