Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fsuppco2 | Structured version Visualization version GIF version |
Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8309 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
fsuppco2.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppco2.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fsuppco2.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐵) |
fsuppco2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
fsuppco2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fsuppco2.n | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppco2.i | ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) |
Ref | Expression |
---|---|
fsuppco2 | ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppco2.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵) | |
2 | ffun 6048 | . . . 4 ⊢ (𝐺:𝐵⟶𝐵 → Fun 𝐺) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
4 | fsuppco2.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
5 | ffun 6048 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
7 | funco 5928 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
8 | 3, 6, 7 | syl2anc 693 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
9 | fsuppco2.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
10 | 9 | fsuppimpd 8282 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
11 | fco 6058 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵) | |
12 | 1, 4, 11 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵) |
13 | eldifi 3732 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
14 | fvco3 6275 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
15 | 4, 13, 14 | syl2an 494 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
16 | ssid 3624 | . . . . . . . 8 ⊢ (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍) | |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
18 | fsuppco2.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
19 | fsuppco2.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
20 | 4, 17, 18, 19 | suppssr 7326 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
21 | 20 | fveq2d 6195 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
22 | fsuppco2.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) | |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍) |
24 | 15, 21, 23 | 3eqtrd 2660 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍) |
25 | 12, 24 | suppss 7325 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
26 | ssfi 8180 | . . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin) | |
27 | 10, 25, 26 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin) |
28 | fsuppco2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
29 | fex 6490 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V) | |
30 | 1, 28, 29 | syl2anc 693 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
31 | fex 6490 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V) | |
32 | 4, 18, 31 | syl2anc 693 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
33 | coexg 7117 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
34 | 30, 32, 33 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
35 | isfsupp 8279 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) | |
36 | 34, 19, 35 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) |
37 | 8, 27, 36 | mpbir2and 957 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 Fincfn 7955 finSupp cfsupp 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-supp 7296 df-er 7742 df-en 7956 df-fin 7959 df-fsupp 8276 |
This theorem is referenced by: gsumzinv 18345 gsumsub 18348 |
Copyright terms: Public domain | W3C validator |