MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsuppcor Structured version   Visualization version   GIF version

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 (𝜑0𝑊)
fsuppcor.z (𝜑𝑍𝐵)
fsuppcor.f (𝜑𝐹:𝐴𝐶)
fsuppcor.g (𝜑𝐺:𝐵𝐷)
fsuppcor.s (𝜑𝐶𝐵)
fsuppcor.a (𝜑𝐴𝑈)
fsuppcor.b (𝜑𝐵𝑉)
fsuppcor.n (𝜑𝐹 finSupp 𝑍)
fsuppcor.i (𝜑 → (𝐺𝑍) = 0 )
Assertion
Ref Expression
fsuppcor (𝜑 → (𝐺𝐹) finSupp 0 )

Proof of Theorem fsuppcor
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsuppcor.g . . . 4 (𝜑𝐺:𝐵𝐷)
2 ffun 6086 . . . 4 (𝐺:𝐵𝐷 → Fun 𝐺)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐺)
4 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
5 ffun 6086 . . . 4 (𝐹:𝐴𝐶 → Fun 𝐹)
64, 5syl 17 . . 3 (𝜑 → Fun 𝐹)
7 funco 5966 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
83, 6, 7syl2anc 694 . 2 (𝜑 → Fun (𝐺𝐹))
9 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
109fsuppimpd 8323 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
121, 11fssresd 6109 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
13 fco2 6097 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1412, 4, 13syl2anc 694 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
15 eldifi 3765 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
16 fvco3 6314 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
174, 15, 16syl2an 493 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
18 ssid 3657 . . . . . . . 8 (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
1918a1i 11 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
20 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
21 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
224, 19, 20, 21suppssr 7371 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2322fveq2d 6233 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
24 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2524adantr 480 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2617, 23, 253eqtrd 2689 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2714, 26suppss 7370 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
28 ssfi 8221 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 0 ) ∈ Fin)
2910, 27, 28syl2anc 694 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
30 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
31 fex 6530 . . . . 5 ((𝐺:𝐵𝐷𝐵𝑉) → 𝐺 ∈ V)
321, 30, 31syl2anc 694 . . . 4 (𝜑𝐺 ∈ V)
33 fex 6530 . . . . 5 ((𝐹:𝐴𝐶𝐴𝑈) → 𝐹 ∈ V)
344, 20, 33syl2anc 694 . . . 4 (𝜑𝐹 ∈ V)
35 coexg 7159 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3632, 34, 35syl2anc 694 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
37 fsuppcor.0 . . 3 (𝜑0𝑊)
38 isfsupp 8320 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3936, 37, 38syl2anc 694 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
408, 29, 39mpbir2and 977 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator