![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > smndex2dlinvh | Structured version Visualization version GIF version |
Description: The halving functions π» are left inverses of the doubling function π·. (Contributed by AV, 18-Feb-2024.) |
Ref | Expression |
---|---|
smndex2dbas.m | β’ π = (EndoFMndββ0) |
smndex2dbas.b | β’ π΅ = (Baseβπ) |
smndex2dbas.0 | β’ 0 = (0gβπ) |
smndex2dbas.d | β’ π· = (π₯ β β0 β¦ (2 Β· π₯)) |
smndex2hbas.n | β’ π β β0 |
smndex2hbas.h | β’ π» = (π₯ β β0 β¦ if(2 β₯ π₯, (π₯ / 2), π)) |
Ref | Expression |
---|---|
smndex2dlinvh | β’ (π» β π·) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12431 | . . 3 β’ 2 β β0 | |
2 | nn0mulcl 12450 | . . . 4 β’ ((2 β β0 β§ π¦ β β0) β (2 Β· π¦) β β0) | |
3 | smndex2dbas.d | . . . . . 6 β’ π· = (π₯ β β0 β¦ (2 Β· π₯)) | |
4 | oveq2 7366 | . . . . . . 7 β’ (π₯ = π¦ β (2 Β· π₯) = (2 Β· π¦)) | |
5 | 4 | cbvmptv 5219 | . . . . . 6 β’ (π₯ β β0 β¦ (2 Β· π₯)) = (π¦ β β0 β¦ (2 Β· π¦)) |
6 | 3, 5 | eqtri 2765 | . . . . 5 β’ π· = (π¦ β β0 β¦ (2 Β· π¦)) |
7 | 6 | a1i 11 | . . . 4 β’ (2 β β0 β π· = (π¦ β β0 β¦ (2 Β· π¦))) |
8 | smndex2hbas.h | . . . . 5 β’ π» = (π₯ β β0 β¦ if(2 β₯ π₯, (π₯ / 2), π)) | |
9 | 8 | a1i 11 | . . . 4 β’ (2 β β0 β π» = (π₯ β β0 β¦ if(2 β₯ π₯, (π₯ / 2), π))) |
10 | breq2 5110 | . . . . 5 β’ (π₯ = (2 Β· π¦) β (2 β₯ π₯ β 2 β₯ (2 Β· π¦))) | |
11 | oveq1 7365 | . . . . 5 β’ (π₯ = (2 Β· π¦) β (π₯ / 2) = ((2 Β· π¦) / 2)) | |
12 | 10, 11 | ifbieq1d 4511 | . . . 4 β’ (π₯ = (2 Β· π¦) β if(2 β₯ π₯, (π₯ / 2), π) = if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π)) |
13 | 2, 7, 9, 12 | fmptco 7076 | . . 3 β’ (2 β β0 β (π» β π·) = (π¦ β β0 β¦ if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π))) |
14 | 1, 13 | ax-mp 5 | . 2 β’ (π» β π·) = (π¦ β β0 β¦ if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π)) |
15 | nn0z 12525 | . . . . . 6 β’ (π¦ β β0 β π¦ β β€) | |
16 | eqidd 2738 | . . . . . 6 β’ (π¦ β β0 β (2 Β· π¦) = (2 Β· π¦)) | |
17 | 2teven 16238 | . . . . . 6 β’ ((π¦ β β€ β§ (2 Β· π¦) = (2 Β· π¦)) β 2 β₯ (2 Β· π¦)) | |
18 | 15, 16, 17 | syl2anc 585 | . . . . 5 β’ (π¦ β β0 β 2 β₯ (2 Β· π¦)) |
19 | 18 | iftrued 4495 | . . . 4 β’ (π¦ β β0 β if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π) = ((2 Β· π¦) / 2)) |
20 | 19 | mpteq2ia 5209 | . . 3 β’ (π¦ β β0 β¦ if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π)) = (π¦ β β0 β¦ ((2 Β· π¦) / 2)) |
21 | nn0cn 12424 | . . . . . 6 β’ (π¦ β β0 β π¦ β β) | |
22 | 2cnd 12232 | . . . . . 6 β’ (π¦ β β0 β 2 β β) | |
23 | 2ne0 12258 | . . . . . . 7 β’ 2 β 0 | |
24 | 23 | a1i 11 | . . . . . 6 β’ (π¦ β β0 β 2 β 0) |
25 | 21, 22, 24 | divcan3d 11937 | . . . . 5 β’ (π¦ β β0 β ((2 Β· π¦) / 2) = π¦) |
26 | 25 | mpteq2ia 5209 | . . . 4 β’ (π¦ β β0 β¦ ((2 Β· π¦) / 2)) = (π¦ β β0 β¦ π¦) |
27 | smndex2dbas.0 | . . . . 5 β’ 0 = (0gβπ) | |
28 | nn0ex 12420 | . . . . . 6 β’ β0 β V | |
29 | smndex2dbas.m | . . . . . . 7 β’ π = (EndoFMndββ0) | |
30 | 29 | efmndid 18699 | . . . . . 6 β’ (β0 β V β ( I βΎ β0) = (0gβπ)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 β’ ( I βΎ β0) = (0gβπ) |
32 | mptresid 6005 | . . . . 5 β’ ( I βΎ β0) = (π¦ β β0 β¦ π¦) | |
33 | 27, 31, 32 | 3eqtr2ri 2772 | . . . 4 β’ (π¦ β β0 β¦ π¦) = 0 |
34 | 26, 33 | eqtri 2765 | . . 3 β’ (π¦ β β0 β¦ ((2 Β· π¦) / 2)) = 0 |
35 | 20, 34 | eqtri 2765 | . 2 β’ (π¦ β β0 β¦ if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π)) = 0 |
36 | 14, 35 | eqtri 2765 | 1 β’ (π» β π·) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β wne 2944 Vcvv 3446 ifcif 4487 class class class wbr 5106 β¦ cmpt 5189 I cid 5531 βΎ cres 5636 β ccom 5638 βcfv 6497 (class class class)co 7358 0cc0 11052 Β· cmul 11057 / cdiv 11813 2c2 12209 β0cn0 12414 β€cz 12500 β₯ cdvds 16137 Basecbs 17084 0gc0g 17322 EndoFMndcefmnd 18679 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-dvds 16138 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-tset 17153 df-0g 17324 df-efmnd 18680 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |