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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e21 | 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 |
---|---|
lmat22e21 | β’ (π β (2π1) = πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . 2 β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) | |
2 | 2nn 12315 | . . 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 14854 | . . 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 14854 | . . 3 β’ (π β β¨βπΆπ·ββ© β Word π) |
10 | 6, 9 | s2cld 14854 | . 2 β’ (π β β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ© β Word Word π) |
11 | s2len 14872 | . . 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 33475 | . 2 β’ ((π β§ π β (0..^2)) β (β―β(β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©βπ)) = 2) |
14 | 1nn0 12518 | . 2 β’ 1 β β0 | |
15 | 0nn0 12517 | . 2 β’ 0 β β0 | |
16 | 2 | nnrei 12251 | . . 3 β’ 2 β β |
17 | 16 | leidi 11778 | . 2 β’ 2 β€ 2 |
18 | 1le2 12451 | . 2 β’ 1 β€ 2 | |
19 | 1p1e2 12367 | . 2 β’ (1 + 1) = 2 | |
20 | 0p1e1 12364 | . 2 β’ (0 + 1) = 1 | |
21 | s2cli 14863 | . . 3 β’ β¨βπΆπ·ββ© β Word V | |
22 | s2fv1 14871 | . . 3 β’ (β¨βπΆπ·ββ© β Word V β (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β1) = β¨βπΆπ·ββ©) | |
23 | 21, 22 | ax-mp 5 | . 2 β’ (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β1) = β¨βπΆπ·ββ© |
24 | s2fv0 14870 | . . 3 β’ (πΆ β π β (β¨βπΆπ·ββ©β0) = πΆ) | |
25 | 7, 24 | syl 17 | . 2 β’ (π β (β¨βπΆπ·ββ©β0) = πΆ) |
26 | 1, 3, 10, 12, 13, 14, 15, 17, 18, 19, 20, 23, 25 | lmatfvlem 33473 | 1 β’ (π β (2π1) = πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3463 βcfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 βcn 12242 2c2 12297 β―chash 14321 Word cword 14496 β¨βcs2 14824 litMatclmat 33469 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-lmat 33470 |
This theorem is referenced by: lmat22det 33480 |
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