Theorem offsplitfpar 8160

Description: Express the function operation map ∘_f by the functions defined in fsplit 8158 and fpar 8157. (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 8159	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
4	3	coeq2d 5887	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
5	4	3ad2ant1 1133	. 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 6759	. . . . . . 7 ⊢ ( + Fn 𝐶 ↔ + :𝐶⟶ran + )
7	6	biimpi 216	. . . . . 6 ⊢ ( + Fn 𝐶 → + :𝐶⟶ran + )
8	7	adantr 480	. . . . 5 ⊢ (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
9	8	3ad2ant3 1135	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
10		simpl3r 1229	. . . . 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)
11		simp1l 1197	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
12		fnfvelrn 7114	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
13	11, 12	sylan 579	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
14		simp1r 1198	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
15		fnfvelrn 7114	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
16	14, 15	sylan 579	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
17	13, 16	opelxpd 5739	. . . . 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))
18	10, 17	sseldd 4009	. . . 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)
19	9, 18	cofmpt 7166	. . 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)⟩)))
20		df-ov 7451	. . . . 5 ⊢ ((𝐹‘𝑎) + (𝐺‘𝑎)) = ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
21	20	eqcomi 2749	. . . 4 ⊢ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = ((𝐹‘𝑎) + (𝐺‘𝑎))
22	21	mpteq2i 5271	. . 3 ⊢ (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
23	19, 22	eqtrdi 2796	. 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))))
24		offval3 8023	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
25		fndm 6682	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
26		fndm 6682	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
27	25, 26	ineqan12d 4243	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
28		inidm 4248	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
29	27, 28	eqtrdi 2796	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
30	29	mpteq1d 5261	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
31	24, 30	sylan9eqr 2802	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
32	31	eqcomd 2746	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
33	32	3adant3 1132	. 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))
34	5, 23, 33	3eqtrd 2784	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ⟨cop 4654   ↦ cmpt 5249   I cid 5592   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-1st 8030  df-2nd 8031

This theorem is referenced by: (None)