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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapvs | Structured version Visualization version Unicode version |
Description: Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hgmapvs.h |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapvs.u |
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hgmapvs.v |
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hgmapvs.t |
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hgmapvs.r |
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hgmapvs.b |
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hgmapvs.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapvs.e |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapvs.s |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapvs.g |
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hgmapvs.k |
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hgmapvs.x |
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hgmapvs.f |
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Ref | Expression |
---|---|
hgmapvs |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapvs.x |
. 2
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2 | hgmapvs.h |
. . . . 5
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3 | hgmapvs.u |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | hgmapvs.v |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | hgmapvs.t |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | hgmapvs.r |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | hgmapvs.b |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | hgmapvs.c |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | hgmapvs.e |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | hgmapvs.s |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | hgmapvs.g |
. . . . 5
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12 | hgmapvs.k |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | hgmapvs.f |
. . . . 5
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14 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | hgmapval 35458 |
. . . 4
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15 | 14 | eqcomd 2457 |
. . 3
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16 | 2, 3, 6, 7, 11, 12, 13 | hgmapcl 35460 |
. . . 4
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17 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 | hdmap14lem15 35453 |
. . . 4
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18 | oveq1 6297 |
. . . . . . 7
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19 | 18 | eqeq2d 2461 |
. . . . . 6
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20 | 19 | ralbidv 2827 |
. . . . 5
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21 | 20 | riota2 6274 |
. . . 4
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22 | 16, 17, 21 | syl2anc 667 |
. . 3
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23 | 15, 22 | mpbird 236 |
. 2
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24 | oveq2 6298 |
. . . . 5
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25 | 24 | fveq2d 5869 |
. . . 4
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26 | fveq2 5865 |
. . . . 5
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27 | 26 | oveq2d 6306 |
. . . 4
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28 | 25, 27 | eqeq12d 2466 |
. . 3
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29 | 28 | rspcva 3148 |
. 2
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30 | 1, 23, 29 | syl2anc 667 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-riotaBAD 32525 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-fal 1450 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-ot 3977 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-of 6531 df-om 6693 df-1st 6793 df-2nd 6794 df-tpos 6973 df-undef 7020 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-n0 10870 df-z 10938 df-uz 11160 df-fz 11785 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-ress 15128 df-plusg 15203 df-mulr 15204 df-sca 15206 df-vsca 15207 df-0g 15340 df-mre 15492 df-mrc 15493 df-acs 15495 df-preset 16173 df-poset 16191 df-plt 16204 df-lub 16220 df-glb 16221 df-join 16222 df-meet 16223 df-p0 16285 df-p1 16286 df-lat 16292 df-clat 16354 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-submnd 16583 df-grp 16673 df-minusg 16674 df-sbg 16675 df-subg 16814 df-cntz 16971 df-oppg 16997 df-lsm 17288 df-cmn 17432 df-abl 17433 df-mgp 17724 df-ur 17736 df-ring 17782 df-oppr 17851 df-dvdsr 17869 df-unit 17870 df-invr 17900 df-dvr 17911 df-drng 17977 df-lmod 18093 df-lss 18156 df-lsp 18195 df-lvec 18326 df-lsatoms 32542 df-lshyp 32543 df-lcv 32585 df-lfl 32624 df-lkr 32652 df-ldual 32690 df-oposet 32742 df-ol 32744 df-oml 32745 df-covers 32832 df-ats 32833 df-atl 32864 df-cvlat 32888 df-hlat 32917 df-llines 33063 df-lplanes 33064 df-lvols 33065 df-lines 33066 df-psubsp 33068 df-pmap 33069 df-padd 33361 df-lhyp 33553 df-laut 33554 df-ldil 33669 df-ltrn 33670 df-trl 33725 df-tgrp 34310 df-tendo 34322 df-edring 34324 df-dveca 34570 df-disoa 34597 df-dvech 34647 df-dib 34707 df-dic 34741 df-dih 34797 df-doch 34916 df-djh 34963 df-lcdual 35155 df-mapd 35193 df-hvmap 35325 df-hdmap1 35362 df-hdmap 35363 df-hgmap 35455 |
This theorem is referenced by: hgmapval0 35463 hgmapval1 35464 hgmapadd 35465 hgmapmul 35466 hgmaprnlem1N 35467 hgmap11 35473 hdmapglnm2 35482 |
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