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Mirrors > Home > MPE Home > Th. List > fzssz | Structured version Visualization version GIF version |
Description: A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fzssz | ⊢ (𝑀...𝑁) ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 12763 | . 2 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
2 | 1 | ssriv 3897 | 1 ⊢ (𝑀...𝑁) ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3863 (class class class)co 7021 ℤcz 11834 ...cfz 12747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-fv 6238 df-ov 7024 df-oprab 7025 df-mpo 7026 df-1st 7550 df-2nd 7551 df-neg 10725 df-z 11835 df-uz 12099 df-fz 12748 |
This theorem is referenced by: fzof 12890 fzossz 12912 seqcoll 13675 prodmolem2a 15126 lcmflefac 15826 prmodvdslcmf 16217 prmolelcmf 16218 prmgaplcmlem1 16221 prmgaplcmlem2 16222 prmgaplcm 16230 wilthlem2 25333 wilthlem3 25334 cycpmfv2 30408 freshmansdream 30518 fsum2dsub 31500 breprexplema 31523 breprexplemc 31525 breprexpnat 31527 vtsprod 31532 circlemeth 31533 fzisoeu 41133 fzsscn 41144 fzssre 41147 fzct 41214 sumnnodd 41478 dvnprodlem1 41798 dvnprodlem2 41799 fourierdlem20 41980 fourierdlem25 41985 fourierdlem37 41997 fourierdlem52 42011 fourierdlem64 42023 fourierdlem79 42038 etransclem32 42119 |
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