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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e22 | Structured version Visualization version GIF version |
Description: Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
Ref | Expression |
---|---|
lmat22.m | β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) |
lmat22.a | β’ (π β π΄ β π) |
lmat22.b | β’ (π β π΅ β π) |
lmat22.c | β’ (π β πΆ β π) |
lmat22.d | β’ (π β π· β π) |
Ref | Expression |
---|---|
lmat22e22 | β’ (π β (2π2) = π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . 2 β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) | |
2 | 2nn 12289 | . . 3 β’ 2 β β | |
3 | 2 | a1i 11 | . 2 β’ (π β 2 β β) |
4 | lmat22.a | . . . 4 β’ (π β π΄ β π) | |
5 | lmat22.b | . . . 4 β’ (π β π΅ β π) | |
6 | 4, 5 | s2cld 14828 | . . 3 β’ (π β β¨βπ΄π΅ββ© β Word π) |
7 | lmat22.c | . . . 4 β’ (π β πΆ β π) | |
8 | lmat22.d | . . . 4 β’ (π β π· β π) | |
9 | 7, 8 | s2cld 14828 | . . 3 β’ (π β β¨βπΆπ·ββ© β Word π) |
10 | 6, 9 | s2cld 14828 | . 2 β’ (π β β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ© β Word Word π) |
11 | s2len 14846 | . . 3 β’ (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2 | |
12 | 11 | a1i 11 | . 2 β’ (π β (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 33327 | . 2 β’ ((π β§ π β (0..^2)) β (β―β(β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©βπ)) = 2) |
14 | 1nn0 12492 | . 2 β’ 1 β β0 | |
15 | 2 | nnrei 12225 | . . 3 β’ 2 β β |
16 | 15 | leidi 11752 | . 2 β’ 2 β€ 2 |
17 | 1p1e2 12341 | . 2 β’ (1 + 1) = 2 | |
18 | s2cli 14837 | . . 3 β’ β¨βπΆπ·ββ© β Word V | |
19 | s2fv1 14845 | . . 3 β’ (β¨βπΆπ·ββ© β Word V β (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β1) = β¨βπΆπ·ββ©) | |
20 | 18, 19 | ax-mp 5 | . 2 β’ (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β1) = β¨βπΆπ·ββ© |
21 | s2fv1 14845 | . . 3 β’ (π· β π β (β¨βπΆπ·ββ©β1) = π·) | |
22 | 8, 21 | syl 17 | . 2 β’ (π β (β¨βπΆπ·ββ©β1) = π·) |
23 | 1, 3, 10, 12, 13, 14, 14, 16, 16, 17, 17, 20, 22 | lmatfvlem 33325 | 1 β’ (π β (2π2) = π·) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 βcfv 6537 (class class class)co 7405 1c1 11113 βcn 12216 2c2 12271 β―chash 14295 Word cword 14470 β¨βcs2 14798 litMatclmat 33321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-lmat 33322 |
This theorem is referenced by: lmat22det 33332 |
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