Theorem fsuppcor 9352

Description: The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppcor.0	⊢ (𝜑 → 0 ∈ 𝑊)
fsuppcor.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
fsuppcor.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
fsuppcor.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
fsuppcor.s	⊢ (𝜑 → 𝐶 ⊆ 𝐵)
fsuppcor.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
fsuppcor.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fsuppcor.n	⊢ (𝜑 → 𝐹 finSupp 𝑍)
fsuppcor.i	⊢ (𝜑 → (𝐺‘𝑍) = 0 )

Assertion

Ref	Expression
fsuppcor	⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Proof of Theorem fsuppcor

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppcor.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
2	1	ffund 6700	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppcor.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffund 6700	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6565	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppcor.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 9317	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fsuppcor.s	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
10	1, 9	fssresd 6735	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷)
11		fco2 6722	. . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
12	10, 3, 11	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
13		eldifi 4087	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 6971	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	3, 13, 14	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssidd 3962	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17		fsuppcor.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
18		fsuppcor.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	3, 16, 17, 18	suppssr 8179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
20	19	fveq2d 6875	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
21		fsuppcor.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 )
22	21	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 )
23	15, 20, 22	3eqtrd 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 )
24	12, 23	suppss 8178	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
25	8, 24	ssfid 9217	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
26		fsuppcor.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
27	1, 26	fexd 7215	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
28	3, 17	fexd 7215	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
29		coexg 7914	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
30	27, 28, 29	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
31		fsuppcor.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑊)
32		isfsupp 9313	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
33	30, 31, 32	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
34	6, 25, 33	mpbir2and 725	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907   class class class wbr 5105   ↾ cres 5654   ∘ ccom 5656  Fun wfun 6519  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   supp csupp 8144  Fincfn 8931   finSupp cfsupp 9309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-supp 8145  df-1o 8441  df-en 8932  df-fin 8935  df-fsupp 9310

This theorem is referenced by:  mapfienlem1  9353  mapfienlem2  9354  mhmcompl  22232  selvvvval  22253  cpmadumatpolylem2  23000  esplympl  33874