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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 47495. (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 12589 | . 2 β’ β€ β V | |
2 | prex 5428 | . 2 β’ {π΄, π΅} β V | |
3 | zlmodzxzldep.a | . . . . . . . 8 β’ π΄ = {β¨0, 3β©, β¨1, 6β©} | |
4 | prex 5428 | . . . . . . . 8 β’ {β¨0, 3β©, β¨1, 6β©} β V | |
5 | 3, 4 | eqeltri 2824 | . . . . . . 7 β’ π΄ β V |
6 | zlmodzxzldep.b | . . . . . . . 8 β’ π΅ = {β¨0, 2β©, β¨1, 4β©} | |
7 | prex 5428 | . . . . . . . 8 β’ {β¨0, 2β©, β¨1, 4β©} β V | |
8 | 6, 7 | eqeltri 2824 | . . . . . . 7 β’ π΅ β V |
9 | 5, 8 | pm3.2i 470 | . . . . . 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 12616 | . . . . . . 7 β’ 2 β β€ | |
12 | 3nn0 12512 | . . . . . . . 8 β’ 3 β β0 | |
13 | 12 | nn0negzi 12623 | . . . . . . 7 β’ -3 β β€ |
14 | 11, 13 | pm3.2i 470 | . . . . . 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 47489 | . . . . . 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 7158 | . . . . . 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 6707 | . . . . . 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 1369 | . . . 4 β’ ((β€ β V β§ {π΄, π΅} β V) β πΉ:{π΄, π΅}βΆ{2, -3}) |
24 | prssi 4820 | . . . . 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 6733 | . . . 4 β’ ((πΉ:{π΄, π΅}βΆ{2, -3} β§ {2, -3} β β€) β πΉ:{π΄, π΅}βΆβ€) | |
27 | 23, 25, 26 | sylancl 585 | . . 3 β’ ((β€ β V β§ {π΄, π΅} β V) β πΉ:{π΄, π΅}βΆβ€) |
28 | elmapg 8849 | . . 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 Vcvv 3469 β wss 3944 {cpr 4626 β¨cop 4630 βΆwf 6538 (class class class)co 7414 βm cmap 8836 0cc0 11130 1c1 11131 -cneg 11467 2c2 12289 3c3 12290 4c4 12291 6c6 12293 β€cz 12580 β€ringczring 21359 freeLMod cfrlm 21667 |
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-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-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-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-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 |
This theorem is referenced by: zlmodzxzldeplem2 47492 zlmodzxzldeplem3 47493 zlmodzxzldep 47495 |
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