fsuppxpfi - Metamath Proof Explorer

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Theorem fsuppxpfi 9375

Description: The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.)

**Assertion**
Ref	Expression
fsuppxpfi	⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin)

**Proof of Theorem fsuppxpfi**
Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐹 finSupp 𝑍 → 𝐹 finSupp 𝑍)
2	1	fsuppimpd 9364	. 2 ⊢ (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin)
3		id 22	. . 3 ⊢ (𝐺 finSupp 𝑍 → 𝐺 finSupp 𝑍)
4	3	fsuppimpd 9364	. 2 ⊢ (𝐺 finSupp 𝑍 → (𝐺 supp 𝑍) ∈ Fin)
5		xpfi 9312	. 2 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ∈ Fin) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin)
6	2, 4, 5	syl2an 595	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 395 ∈ wcel 2098 class class class wbr 5138 × cxp 5664 (class class class)co 7401 supp csupp 8140 Fincfn 8934 finSupp cfsupp 9356

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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718

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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-om 7849 df-1o 8461 df-en 8935 df-fin 8938 df-fsupp 9357

This theorem is referenced by: mplsubrglem 21872

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