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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 12511 | . . 3 β’ 2 β β0 | |
2 | nn0mulcl 12530 | . . . 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 7422 | . . . . . . 7 β’ (π₯ = π¦ β (2 Β· π₯) = (2 Β· π¦)) | |
5 | 4 | cbvmptv 5255 | . . . . . 6 β’ (π₯ β β0 β¦ (2 Β· π₯)) = (π¦ β β0 β¦ (2 Β· π¦)) |
6 | 3, 5 | eqtri 2755 | . . . . 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 5146 | . . . . 5 β’ (π₯ = (2 Β· π¦) β (2 β₯ π₯ β 2 β₯ (2 Β· π¦))) | |
11 | oveq1 7421 | . . . . 5 β’ (π₯ = (2 Β· π¦) β (π₯ / 2) = ((2 Β· π¦) / 2)) | |
12 | 10, 11 | ifbieq1d 4548 | . . . 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 7132 | . . 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 12605 | . . . . . 6 β’ (π¦ β β0 β π¦ β β€) | |
16 | eqidd 2728 | . . . . . 6 β’ (π¦ β β0 β (2 Β· π¦) = (2 Β· π¦)) | |
17 | 2teven 16323 | . . . . . 6 β’ ((π¦ β β€ β§ (2 Β· π¦) = (2 Β· π¦)) β 2 β₯ (2 Β· π¦)) | |
18 | 15, 16, 17 | syl2anc 583 | . . . . 5 β’ (π¦ β β0 β 2 β₯ (2 Β· π¦)) |
19 | 18 | iftrued 4532 | . . . 4 β’ (π¦ β β0 β if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π) = ((2 Β· π¦) / 2)) |
20 | 19 | mpteq2ia 5245 | . . 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 12504 | . . . . . 6 β’ (π¦ β β0 β π¦ β β) | |
22 | 2cnd 12312 | . . . . . 6 β’ (π¦ β β0 β 2 β β) | |
23 | 2ne0 12338 | . . . . . . 7 β’ 2 β 0 | |
24 | 23 | a1i 11 | . . . . . 6 β’ (π¦ β β0 β 2 β 0) |
25 | 21, 22, 24 | divcan3d 12017 | . . . . 5 β’ (π¦ β β0 β ((2 Β· π¦) / 2) = π¦) |
26 | 25 | mpteq2ia 5245 | . . . 4 β’ (π¦ β β0 β¦ ((2 Β· π¦) / 2)) = (π¦ β β0 β¦ π¦) |
27 | smndex2dbas.0 | . . . . 5 β’ 0 = (0gβπ) | |
28 | nn0ex 12500 | . . . . . 6 β’ β0 β V | |
29 | smndex2dbas.m | . . . . . . 7 β’ π = (EndoFMndββ0) | |
30 | 29 | efmndid 18831 | . . . . . 6 β’ (β0 β V β ( I βΎ β0) = (0gβπ)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 β’ ( I βΎ β0) = (0gβπ) |
32 | mptresid 6048 | . . . . 5 β’ ( I βΎ β0) = (π¦ β β0 β¦ π¦) | |
33 | 27, 31, 32 | 3eqtr2ri 2762 | . . . 4 β’ (π¦ β β0 β¦ π¦) = 0 |
34 | 26, 33 | eqtri 2755 | . . 3 β’ (π¦ β β0 β¦ ((2 Β· π¦) / 2)) = 0 |
35 | 20, 34 | eqtri 2755 | . 2 β’ (π¦ β β0 β¦ if(2 β₯ (2 Β· π¦), ((2 Β· π¦) / 2), π)) = 0 |
36 | 14, 35 | eqtri 2755 | 1 β’ (π» β π·) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 β wne 2935 Vcvv 3469 ifcif 4524 class class class wbr 5142 β¦ cmpt 5225 I cid 5569 βΎ cres 5674 β ccom 5676 βcfv 6542 (class class class)co 7414 0cc0 11130 Β· cmul 11135 / cdiv 11893 2c2 12289 β0cn0 12494 β€cz 12580 β₯ cdvds 16222 Basecbs 17171 0gc0g 17412 EndoFMndcefmnd 18811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-dvds 16223 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-tset 17243 df-0g 17414 df-efmnd 18812 |
This theorem is referenced by: (None) |
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