Theorem fsuppcor 8350

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		ffun 6086	. . . 4 ⊢ (𝐺:𝐵⟶𝐷 → Fun 𝐺)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐺)
4		fsuppcor.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
5		ffun 6086	. . . 4 ⊢ (𝐹:𝐴⟶𝐶 → Fun 𝐹)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
7		funco 5966	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
8	3, 6, 7	syl2anc 694	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
9		fsuppcor.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
10	9	fsuppimpd 8323	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11		fsuppcor.s	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
12	1, 11	fssresd 6109	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷)
13		fco2 6097	. . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
14	12, 4, 13	syl2anc 694	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
15		eldifi 3765	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
16		fvco3 6314	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
17	4, 15, 16	syl2an 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
18		ssid 3657	. . . . . . . 8 ⊢ (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
20		fsuppcor.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
21		fsuppcor.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22	4, 19, 20, 21	suppssr 7371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
23	22	fveq2d 6233	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
24		fsuppcor.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 )
25	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 )
26	17, 23, 25	3eqtrd 2689	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 )
27	14, 26	suppss 7370	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
28		ssfi 8221	. . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
29	10, 27, 28	syl2anc 694	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
30		fsuppcor.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
31		fex 6530	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐷 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
32	1, 30, 31	syl2anc 694	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
33		fex 6530	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V)
34	4, 20, 33	syl2anc 694	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
35		coexg 7159	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
36	32, 34, 35	syl2anc 694	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
37		fsuppcor.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑊)
38		isfsupp 8320	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
39	36, 37, 38	syl2anc 694	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
40	8, 29, 39	mpbir2and 977	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607   class class class wbr 4685   ↾ cres 5145   ∘ ccom 5147  Fun wfun 5920  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   supp csupp 7340  Fincfn 7997   finSupp cfsupp 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-supp 7341  df-er 7787  df-en 7998  df-fin 8001  df-fsupp 8317

This theorem is referenced by:  mapfienlem1  8351  mapfienlem2  8352  cpmadumatpolylem2  20735