Theorem fsuppcor 9473

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 6751	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppcor.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffund 6751	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6618	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 583	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppcor.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 9439	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fsuppcor.s	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
10	1, 9	fssresd 6788	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷)
11		fco2 6774	. . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
12	10, 3, 11	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
13		eldifi 4154	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 7021	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	3, 13, 14	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssidd 4032	. . . . . . 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 8236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
20	19	fveq2d 6924	. . . . 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 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 )
24	12, 23	suppss 8235	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
25	8, 24	ssfid 9329	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
26		fsuppcor.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
27	1, 26	fexd 7264	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
28	3, 17	fexd 7264	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
29		coexg 7969	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
30	27, 28, 29	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
31		fsuppcor.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑊)
32		isfsupp 9435	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
33	30, 31, 32	syl2anc 583	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
34	6, 25, 33	mpbir2and 712	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976   class class class wbr 5166   ↾ cres 5702   ∘ ccom 5704  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   supp csupp 8201  Fincfn 9003   finSupp cfsupp 9431

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-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  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-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  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-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  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-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  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-om 7904  df-supp 8202  df-1o 8522  df-en 9004  df-fin 9007  df-fsupp 9432

This theorem is referenced by:  mapfienlem1  9474  mapfienlem2  9475  mhmcompl  22405  cpmadumatpolylem2  22909  selvvvval  42540