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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 9370 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | fsuppun.g | . . . . 5 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
4 | fsuppimp 9370 | . . . . 5 ⊢ (𝐺 finSupp 𝑍 → (Fun 𝐺 ∧ (𝐺 supp 𝑍) ∈ Fin)) | |
5 | unfi 9174 | . . . . . . 7 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ∈ Fin) → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin) | |
6 | 5 | expcom 413 | . . . . . 6 ⊢ ((𝐺 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)) |
7 | 6 | adantl 481 | . . . . 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 503 | . 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 395 ∈ wcel 2098 ∪ cun 3941 class class class wbr 5141 Fun wfun 6531 (class class class)co 7405 supp csupp 8146 Fincfn 8941 finSupp cfsupp 9363 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-om 7853 df-en 8942 df-fin 8945 df-fsupp 9364 |
This theorem is referenced by: wemapso2lem 9549 dprdfadd 19942 psrbagaddcl 21822 mhpmulcl 22032 naddcnff 42688 |
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