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Mirrors > Home > MPE Home > Th. List > fv0p1e1 | Structured version Visualization version GIF version |
Description: Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
Ref | Expression |
---|---|
fv0p1e1 | ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7278 | . . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
2 | 0p1e1 12095 | . . 3 ⊢ (0 + 1) = 1 | |
3 | 1, 2 | eqtrdi 2796 | . 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
4 | 3 | fveq2d 6775 | 1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ‘cfv 6432 (class class class)co 7271 0cc0 10872 1c1 10873 + caddc 10875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-ltxr 11015 |
This theorem is referenced by: mertenslem2 15595 loglesqrt 25909 harmonicbnd3 26155 facgam 26213 wlkonl1iedg 28030 2wlklem 28032 pthdadjvtx 28094 lfgrn1cycl 28166 0enwwlksnge1 28225 2wlkdlem5 28290 2wlkdlem10 28296 rusgrnumwwlkl1 28329 clwwlkn2 28404 3wlkdlem5 28523 3wlkdlem10 28529 upgr3v3e3cycl 28540 upgr4cycl4dv4e 28545 subfacval2 33145 iccelpart 44854 bgoldbtbnd 45230 |
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