![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsuppco | Structured version Visualization version GIF version |
Description: The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.) |
Ref | Expression |
---|---|
fsuppco.f | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppco.g | ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) |
fsuppco.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppco.v | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
fsuppco | ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppco.v | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | fsuppco.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
3 | df-f1 6558 | . . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
4 | 3 | simprbi 495 | . . . . . 6 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun ◡𝐺) |
6 | cofunex2g 7959 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V) | |
7 | 1, 5, 6 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
8 | fsuppco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
9 | suppimacnv 8185 | . . . 4 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) | |
10 | 7, 8, 9 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) |
11 | suppimacnv 8185 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
12 | 1, 8, 11 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
13 | fsuppco.f | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
14 | 13 | fsuppimpd 9401 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
15 | 12, 14 | eqeltrrd 2830 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
16 | 15, 2 | fsuppcolem 9432 | . . 3 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
17 | 10, 16 | eqeltrd 2829 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin) |
18 | fsuppimp 9400 | . . . . . 6 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
19 | 18 | simpld 493 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
21 | f1fun 6800 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐺) |
23 | funco 6598 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
24 | 20, 22, 23 | syl2anc 582 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
25 | funisfsupp 9399 | . . 3 ⊢ ((Fun (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) | |
26 | 24, 7, 8, 25 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) |
27 | 17, 26 | mpbird 256 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∖ cdif 3946 {csn 4632 class class class wbr 5152 ◡ccnv 5681 “ cima 5685 ∘ ccom 5686 Fun wfun 6547 ⟶wf 6549 –1-1→wf1 6550 (class class class)co 7426 supp csupp 8171 Fincfn 8970 finSupp cfsupp 9393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-supp 8172 df-1o 8493 df-en 8971 df-fin 8974 df-fsupp 9394 |
This theorem is referenced by: mapfienlem1 9436 mapfienlem2 9437 psdmplcl 22093 coe1sfi 22139 gsumpart 32790 evlselv 41851 |
Copyright terms: Public domain | W3C validator |