Theorem offsplitfpar 8110

Description: Express the function operation map ∘_f by the functions defined in fsplit 8108 and fpar 8107. (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 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Proof of Theorem offsplitfpar

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	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)
3	1, 2	fsplitfpar 8109	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
4	3	coeq2d 5848	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
5	4	3ad2ant1 1151	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
6		dffn3 6718	. . . . . 6 ⊢ ( + Fn 𝐶 ↔ + :𝐶⟶ran + )
7	6	birani 508	. . . . 5 ⊢ (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
8	7	3ad2ant3 1153	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
9		simpl3r 1248	. . . . 5 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (ran 𝐹 × ran 𝐺) ⊆ 𝐶)
10		simp1l 1216	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
11		fnfvelrn 7075	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
12	10, 11	sylan 591	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
13		simp1r 1217	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
14		fnfvelrn 7075	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
15	13, 14	sylan 591	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
16	12, 15	opelxpd 5700	. . . . 5 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (ran 𝐹 × ran 𝐺))
17	9, 16	sseldd 3938	. . . 4 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ 𝐶)
18	8, 17	cofmpt 7128	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
19		df-ov 7413	. . . . 5 ⊢ ((𝐹‘𝑎) + (𝐺‘𝑎)) = ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
20	19	eqcomi 2772	. . . 4 ⊢ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = ((𝐹‘𝑎) + (𝐺‘𝑎))
21	20	mpteq2i 5207	. . 3 ⊢ (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	18, 21	eqtrdi 2814	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
23		offval3 7975	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
24		fndm 6638	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
25		fndm 6638	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
26	24, 25	ineqan12d 4175	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
27		inidm 4179	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
28	26, 27	eqtrdi 2814	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
29	28	mpteq1d 5201	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
30	23, 29	sylan9eqr 2820	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
31	30	eqcomd 2769	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
32	31	3adant3 1150	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
33	5, 22, 32	3eqtrd 2802	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4595   ↦ cmpt 5192   I cid 5555   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   ∘ ccom 5665   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672  1^st c1st 7980  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-1st 7982  df-2nd 7983

This theorem is referenced by: (None)