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Mirrors > Home > MPE Home > Th. List > fsuppxpfi | Structured version Visualization version GIF version |
Description: The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.) |
Ref | Expression |
---|---|
fsuppxpfi | ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐹 finSupp 𝑍 → 𝐹 finSupp 𝑍) | |
2 | 1 | fsuppimpd 9246 | . 2 ⊢ (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin) |
3 | id 22 | . . 3 ⊢ (𝐺 finSupp 𝑍 → 𝐺 finSupp 𝑍) | |
4 | 3 | fsuppimpd 9246 | . 2 ⊢ (𝐺 finSupp 𝑍 → (𝐺 supp 𝑍) ∈ Fin) |
5 | xpfi 9195 | . 2 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ∈ Fin) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin) | |
6 | 2, 4, 5 | syl2an 597 | 1 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5104 × cxp 5629 (class class class)co 7350 supp csupp 8060 Fincfn 8817 finSupp cfsupp 9239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-om 7794 df-1o 8380 df-en 8818 df-fin 8821 df-fsupp 9240 |
This theorem is referenced by: mplsubrglem 21338 |
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