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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 46671. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxzldep.z | β’ π = (β€ring freeLMod {0, 1}) |
zlmodzxzldep.a | β’ π΄ = {β¨0, 3β©, β¨1, 6β©} |
zlmodzxzldep.b | β’ π΅ = {β¨0, 2β©, β¨1, 4β©} |
zlmodzxzldeplem.f | β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} |
Ref | Expression |
---|---|
zlmodzxzldeplem1 | β’ πΉ β (β€ βm {π΄, π΅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 12513 | . 2 β’ β€ β V | |
2 | prex 5390 | . 2 β’ {π΄, π΅} β V | |
3 | zlmodzxzldep.a | . . . . . . . 8 β’ π΄ = {β¨0, 3β©, β¨1, 6β©} | |
4 | prex 5390 | . . . . . . . 8 β’ {β¨0, 3β©, β¨1, 6β©} β V | |
5 | 3, 4 | eqeltri 2830 | . . . . . . 7 β’ π΄ β V |
6 | zlmodzxzldep.b | . . . . . . . 8 β’ π΅ = {β¨0, 2β©, β¨1, 4β©} | |
7 | prex 5390 | . . . . . . . 8 β’ {β¨0, 2β©, β¨1, 4β©} β V | |
8 | 6, 7 | eqeltri 2830 | . . . . . . 7 β’ π΅ β V |
9 | 5, 8 | pm3.2i 472 | . . . . . 6 β’ (π΄ β V β§ π΅ β V) |
10 | 9 | a1i 11 | . . . . 5 β’ ((β€ β V β§ {π΄, π΅} β V) β (π΄ β V β§ π΅ β V)) |
11 | 2z 12540 | . . . . . . 7 β’ 2 β β€ | |
12 | 3nn0 12436 | . . . . . . . 8 β’ 3 β β0 | |
13 | 12 | nn0negzi 12547 | . . . . . . 7 β’ -3 β β€ |
14 | 11, 13 | pm3.2i 472 | . . . . . 6 β’ (2 β β€ β§ -3 β β€) |
15 | 14 | a1i 11 | . . . . 5 β’ ((β€ β V β§ {π΄, π΅} β V) β (2 β β€ β§ -3 β β€)) |
16 | zlmodzxzldep.z | . . . . . . 7 β’ π = (β€ring freeLMod {0, 1}) | |
17 | 16, 3, 6 | zlmodzxzldeplem 46665 | . . . . . 6 β’ π΄ β π΅ |
18 | 17 | a1i 11 | . . . . 5 β’ ((β€ β V β§ {π΄, π΅} β V) β π΄ β π΅) |
19 | fprg 7102 | . . . . . 6 β’ (((π΄ β V β§ π΅ β V) β§ (2 β β€ β§ -3 β β€) β§ π΄ β π΅) β {β¨π΄, 2β©, β¨π΅, -3β©}:{π΄, π΅}βΆ{2, -3}) | |
20 | zlmodzxzldeplem.f | . . . . . . 7 β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} | |
21 | 20 | feq1i 6660 | . . . . . 6 β’ (πΉ:{π΄, π΅}βΆ{2, -3} β {β¨π΄, 2β©, β¨π΅, -3β©}:{π΄, π΅}βΆ{2, -3}) |
22 | 19, 21 | sylibr 233 | . . . . 5 β’ (((π΄ β V β§ π΅ β V) β§ (2 β β€ β§ -3 β β€) β§ π΄ β π΅) β πΉ:{π΄, π΅}βΆ{2, -3}) |
23 | 10, 15, 18, 22 | syl3anc 1372 | . . . 4 β’ ((β€ β V β§ {π΄, π΅} β V) β πΉ:{π΄, π΅}βΆ{2, -3}) |
24 | prssi 4782 | . . . . 5 β’ ((2 β β€ β§ -3 β β€) β {2, -3} β β€) | |
25 | 11, 13, 24 | mp2an 691 | . . . 4 β’ {2, -3} β β€ |
26 | fss 6686 | . . . 4 β’ ((πΉ:{π΄, π΅}βΆ{2, -3} β§ {2, -3} β β€) β πΉ:{π΄, π΅}βΆβ€) | |
27 | 23, 25, 26 | sylancl 587 | . . 3 β’ ((β€ β V β§ {π΄, π΅} β V) β πΉ:{π΄, π΅}βΆβ€) |
28 | elmapg 8781 | . . 3 β’ ((β€ β V β§ {π΄, π΅} β V) β (πΉ β (β€ βm {π΄, π΅}) β πΉ:{π΄, π΅}βΆβ€)) | |
29 | 27, 28 | mpbird 257 | . 2 β’ ((β€ β V β§ {π΄, π΅} β V) β πΉ β (β€ βm {π΄, π΅})) |
30 | 1, 2, 29 | mp2an 691 | 1 β’ πΉ β (β€ βm {π΄, π΅}) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3444 β wss 3911 {cpr 4589 β¨cop 4593 βΆwf 6493 (class class class)co 7358 βm cmap 8768 0cc0 11056 1c1 11057 -cneg 11391 2c2 12213 3c3 12214 4c4 12215 6c6 12217 β€cz 12504 β€ringczring 20885 freeLMod cfrlm 21168 |
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-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-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-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 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-3 12222 df-n0 12419 df-z 12505 |
This theorem is referenced by: zlmodzxzldeplem2 46668 zlmodzxzldeplem3 46669 zlmodzxzldep 46671 |
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