Theorem fsuppcor 9442

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 6741	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppcor.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffund 6741	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6608	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 584	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppcor.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 9407	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fsuppcor.s	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
10	1, 9	fssresd 6776	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷)
11		fco2 6763	. . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
12	10, 3, 11	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
13		eldifi 4141	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 7008	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	3, 13, 14	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssidd 4019	. . . . . . 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 8219	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
20	19	fveq2d 6911	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
21		fsuppcor.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 )
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 )
23	15, 20, 22	3eqtrd 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 )
24	12, 23	suppss 8218	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
25	8, 24	ssfid 9299	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
26		fsuppcor.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
27	1, 26	fexd 7247	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
28	3, 17	fexd 7247	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
29		coexg 7952	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
30	27, 28, 29	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
31		fsuppcor.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑊)
32		isfsupp 9403	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
33	30, 31, 32	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
34	6, 25, 33	mpbir2and 713	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963   class class class wbr 5148   ↾ cres 5691   ∘ ccom 5693  Fun wfun 6557  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   supp csupp 8184  Fincfn 8984   finSupp cfsupp 9399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-supp 8185  df-1o 8505  df-en 8985  df-fin 8988  df-fsupp 9400

This theorem is referenced by:  mapfienlem1  9443  mapfienlem2  9444  mhmcompl  22400  cpmadumatpolylem2  22904  selvvvval  42572