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Mirrors > Home > ILE Home > Th. List > lspsneq0b | GIF version |
Description: Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.) |
Ref | Expression |
---|---|
lspsneq0b.v | β’ π = (Baseβπ) |
lspsneq0b.o | β’ 0 = (0gβπ) |
lspsneq0b.n | β’ π = (LSpanβπ) |
lspsneq0b.w | β’ (π β π β LMod) |
lspsneq0b.x | β’ (π β π β π) |
lspsneq0b.y | β’ (π β π β π) |
lspsneq0b.e | β’ (π β (πβ{π}) = (πβ{π})) |
Ref | Expression |
---|---|
lspsneq0b | β’ (π β (π = 0 β π = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsneq0b.e | . . . . 5 β’ (π β (πβ{π}) = (πβ{π})) | |
2 | 1 | adantr 276 | . . . 4 β’ ((π β§ π = 0 ) β (πβ{π}) = (πβ{π})) |
3 | lspsneq0b.w | . . . . . 6 β’ (π β π β LMod) | |
4 | lspsneq0b.x | . . . . . 6 β’ (π β π β π) | |
5 | lspsneq0b.v | . . . . . . 7 β’ π = (Baseβπ) | |
6 | lspsneq0b.o | . . . . . . 7 β’ 0 = (0gβπ) | |
7 | lspsneq0b.n | . . . . . . 7 β’ π = (LSpanβπ) | |
8 | 5, 6, 7 | lspsneq0 13703 | . . . . . 6 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
9 | 3, 4, 8 | syl2anc 411 | . . . . 5 β’ (π β ((πβ{π}) = { 0 } β π = 0 )) |
10 | 9 | biimpar 297 | . . . 4 β’ ((π β§ π = 0 ) β (πβ{π}) = { 0 }) |
11 | 2, 10 | eqtr3d 2224 | . . 3 β’ ((π β§ π = 0 ) β (πβ{π}) = { 0 }) |
12 | lspsneq0b.y | . . . . 5 β’ (π β π β π) | |
13 | 5, 6, 7 | lspsneq0 13703 | . . . . 5 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
14 | 3, 12, 13 | syl2anc 411 | . . . 4 β’ (π β ((πβ{π}) = { 0 } β π = 0 )) |
15 | 14 | adantr 276 | . . 3 β’ ((π β§ π = 0 ) β ((πβ{π}) = { 0 } β π = 0 )) |
16 | 11, 15 | mpbid 147 | . 2 β’ ((π β§ π = 0 ) β π = 0 ) |
17 | 1 | adantr 276 | . . . 4 β’ ((π β§ π = 0 ) β (πβ{π}) = (πβ{π})) |
18 | 14 | biimpar 297 | . . . 4 β’ ((π β§ π = 0 ) β (πβ{π}) = { 0 }) |
19 | 17, 18 | eqtrd 2222 | . . 3 β’ ((π β§ π = 0 ) β (πβ{π}) = { 0 }) |
20 | 9 | adantr 276 | . . 3 β’ ((π β§ π = 0 ) β ((πβ{π}) = { 0 } β π = 0 )) |
21 | 19, 20 | mpbid 147 | . 2 β’ ((π β§ π = 0 ) β π = 0 ) |
22 | 16, 21 | impbida 596 | 1 β’ (π β (π = 0 β π = 0 )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1364 β wcel 2160 {csn 3607 βcfv 5231 Basecbs 12480 0gc0g 12727 LModclmod 13564 LSpanclspn 13663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-mgp 13236 df-ring 13313 df-lmod 13566 df-lssm 13630 df-lsp 13664 |
This theorem is referenced by: (None) |
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