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Mirrors > Home > MPE Home > Th. List > lspsnvsi | Structured version Visualization version Unicode version |
Description: Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.) |
Ref | Expression |
---|---|
lspsn.f |
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lspsn.k |
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lspsn.v |
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lspsn.t |
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lspsn.n |
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Ref | Expression |
---|---|
lspsnvsi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2450 |
. 2
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2 | lspsn.n |
. 2
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3 | simp1 1007 |
. 2
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4 | simp3 1009 |
. . . 4
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5 | 4 | snssd 4116 |
. . 3
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6 | lspsn.v |
. . . 4
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7 | 6, 1, 2 | lspcl 18192 |
. . 3
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8 | 3, 5, 7 | syl2anc 666 |
. 2
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9 | lspsn.t |
. . 3
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10 | lspsn.f |
. . 3
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11 | lspsn.k |
. . 3
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12 | simp2 1008 |
. . 3
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13 | 6, 9, 10, 11, 2, 3, 12, 4 | lspsneli 18217 |
. 2
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14 | 1, 2, 3, 8, 13 | lspsnel5a 18212 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-nn 10607 df-2 10665 df-ndx 15117 df-slot 15118 df-base 15119 df-sets 15120 df-plusg 15196 df-0g 15333 df-mgm 16481 df-sgrp 16520 df-mnd 16530 df-grp 16666 df-minusg 16667 df-sbg 16668 df-mgp 17717 df-ur 17729 df-ring 17775 df-lmod 18086 df-lss 18149 df-lsp 18188 |
This theorem is referenced by: lspsnneg 18222 lspsnvs 18330 lclkrlem2p 35084 hgmaprnlem2N 35462 |
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