![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsuppcor | Structured version Visualization version GIF version |
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.) |
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 ) |
Ref | Expression |
---|---|
fsuppcor | ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppcor.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) | |
2 | 1 | ffund 6345 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppcor.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
4 | 3 | ffund 6345 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6225 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 576 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppcor.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 8633 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fsuppcor.s | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
10 | 1, 9 | fssresd 6371 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷) |
11 | fco2 6359 | . . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷) | |
12 | 10, 3, 11 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷) |
13 | eldifi 3987 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
14 | fvco3 6586 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
15 | 3, 13, 14 | syl2an 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
16 | ssidd 3874 | . . . . . . 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 7662 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
20 | 19 | fveq2d 6500 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
21 | fsuppcor.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) | |
22 | 21 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 ) |
23 | 15, 20, 22 | 3eqtrd 2812 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 ) |
24 | 12, 23 | suppss 7661 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) |
25 | 8, 24 | ssfid 8534 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin) |
26 | fsuppcor.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
27 | fex 6813 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐷 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V) | |
28 | 1, 26, 27 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
29 | fex 6813 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V) | |
30 | 3, 17, 29 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
31 | coexg 7447 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
32 | 28, 30, 31 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
33 | fsuppcor.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑊) | |
34 | isfsupp 8630 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) | |
35 | 32, 33, 34 | syl2anc 576 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) |
36 | 6, 25, 35 | mpbir2and 700 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ∖ cdif 3820 ⊆ wss 3823 class class class wbr 4925 ↾ cres 5405 ∘ ccom 5407 Fun wfun 6179 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 supp csupp 7631 Fincfn 8304 finSupp cfsupp 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-supp 7632 df-er 8087 df-en 8305 df-fin 8308 df-fsupp 8627 |
This theorem is referenced by: mapfienlem1 8661 mapfienlem2 8662 cpmadumatpolylem2 21206 |
Copyright terms: Public domain | W3C validator |