Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7275 | . . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
2 | 0p1e1 12078 | . . 3 ⊢ (0 + 1) = 1 | |
3 | 1, 2 | eqtrdi 2795 | . 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
4 | 3 | fveq2d 6772 | 1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ‘cfv 6430 (class class class)co 7268 0cc0 10855 1c1 10856 + caddc 10858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-ltxr 10998 |
This theorem is referenced by: mertenslem2 15578 loglesqrt 25892 harmonicbnd3 26138 facgam 26196 wlkonl1iedg 28013 2wlklem 28015 pthdadjvtx 28077 lfgrn1cycl 28149 0enwwlksnge1 28208 2wlkdlem5 28273 2wlkdlem10 28279 rusgrnumwwlkl1 28312 clwwlkn2 28387 3wlkdlem5 28506 3wlkdlem10 28512 upgr3v3e3cycl 28523 upgr4cycl4dv4e 28528 subfacval2 33128 iccelpart 44837 bgoldbtbnd 45213 |
Copyright terms: Public domain | W3C validator |