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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e11 | Structured version Visualization version GIF version |
Description: Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-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 |
---|---|
lmat22e11 | β’ (π β (1π1) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . . 3 β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) | |
2 | 2nn 12231 | . . . 4 β’ 2 β β | |
3 | 2 | a1i 11 | . . 3 β’ (π β 2 β β) |
4 | lmat22.a | . . . . 5 β’ (π β π΄ β π) | |
5 | lmat22.b | . . . . 5 β’ (π β π΅ β π) | |
6 | 4, 5 | s2cld 14766 | . . . 4 β’ (π β β¨βπ΄π΅ββ© β Word π) |
7 | lmat22.c | . . . . 5 β’ (π β πΆ β π) | |
8 | lmat22.d | . . . . 5 β’ (π β π· β π) | |
9 | 7, 8 | s2cld 14766 | . . . 4 β’ (π β β¨βπΆπ·ββ© β Word π) |
10 | 6, 9 | s2cld 14766 | . . 3 β’ (π β β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ© β Word Word π) |
11 | s2len 14784 | . . . 4 β’ (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2 | |
12 | 11 | a1i 11 | . . 3 β’ (π β (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 32455 | . . 3 β’ ((π β§ π β (0..^2)) β (β―β(β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©βπ)) = 2) |
14 | 2eluzge1 12824 | . . . . 5 β’ 2 β (β€β₯β1) | |
15 | eluzfz1 13454 | . . . . 5 β’ (2 β (β€β₯β1) β 1 β (1...2)) | |
16 | 14, 15 | ax-mp 5 | . . . 4 β’ 1 β (1...2) |
17 | 16 | a1i 11 | . . 3 β’ (π β 1 β (1...2)) |
18 | 1, 3, 10, 12, 13, 17, 17 | lmatfval 32452 | . 2 β’ (π β (1π1) = ((β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β(1 β 1))β(1 β 1))) |
19 | 1m1e0 12230 | . . . . 5 β’ (1 β 1) = 0 | |
20 | 19 | fveq2i 6846 | . . . 4 β’ (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β(1 β 1)) = (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β0) |
21 | s2fv0 14782 | . . . . 5 β’ (β¨βπ΄π΅ββ© β Word π β (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β0) = β¨βπ΄π΅ββ©) | |
22 | 6, 21 | syl 17 | . . . 4 β’ (π β (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β0) = β¨βπ΄π΅ββ©) |
23 | 20, 22 | eqtrid 2785 | . . 3 β’ (π β (β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β(1 β 1)) = β¨βπ΄π΅ββ©) |
24 | 19 | a1i 11 | . . 3 β’ (π β (1 β 1) = 0) |
25 | 23, 24 | fveq12d 6850 | . 2 β’ (π β ((β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©β(1 β 1))β(1 β 1)) = (β¨βπ΄π΅ββ©β0)) |
26 | s2fv0 14782 | . . 3 β’ (π΄ β π β (β¨βπ΄π΅ββ©β0) = π΄) | |
27 | 4, 26 | syl 17 | . 2 β’ (π β (β¨βπ΄π΅ββ©β0) = π΄) |
28 | 18, 25, 27 | 3eqtrd 2777 | 1 β’ (π β (1π1) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 β cmin 11390 βcn 12158 2c2 12213 β€β₯cuz 12768 ...cfz 13430 β―chash 14236 Word cword 14408 β¨βcs2 14736 litMatclmat 32449 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 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-op 4594 df-uni 4867 df-int 4909 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 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-s2 14743 df-lmat 32450 |
This theorem is referenced by: lmat22det 32460 |
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