Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fsuppunfi | Structured version Visualization version GIF version |
Description: The union of the support of two finitely supported functions is finite. (Contributed by AV, 1-Jul-2019.) |
Ref | Expression |
---|---|
fsuppun.f | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppun.g | ⊢ (𝜑 → 𝐺 finSupp 𝑍) |
Ref | Expression |
---|---|
fsuppunfi | ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppun.f | . 2 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
2 | fsuppimp 8839 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | fsuppun.g | . . . . 5 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
4 | fsuppimp 8839 | . . . . 5 ⊢ (𝐺 finSupp 𝑍 → (Fun 𝐺 ∧ (𝐺 supp 𝑍) ∈ Fin)) | |
5 | unfi 8785 | . . . . . . 7 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ∈ Fin) → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin) | |
6 | 5 | expcom 416 | . . . . . 6 ⊢ ((𝐺 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
7 | 6 | adantl 484 | . . . . 5 ⊢ ((Fun 𝐺 ∧ (𝐺 supp 𝑍) ∈ Fin) → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
8 | 3, 4, 7 | 3syl 18 | . . . 4 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
9 | 8 | com12 32 | . . 3 ⊢ ((𝐹 supp 𝑍) ∈ Fin → (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
10 | 2, 9 | simpl2im 506 | . 2 ⊢ (𝐹 finSupp 𝑍 → (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
11 | 1, 10 | mpcom 38 | 1 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∪ cun 3934 class class class wbr 5066 Fun wfun 6349 (class class class)co 7156 supp csupp 7830 Fincfn 8509 finSupp cfsupp 8833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fsupp 8834 |
This theorem is referenced by: wemapso2lem 9016 dprdfadd 19142 |
Copyright terms: Public domain | W3C validator |