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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 13579 | . . . . . 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 2222 | . . 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 13579 | . . . . 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 2220 | . . 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 1363 β wcel 2158 {csn 3604 βcfv 5228 Basecbs 12475 0gc0g 12722 LModclmod 13440 LSpanclspn 13539 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-plusg 12563 df-mulr 12564 df-sca 12566 df-vsca 12567 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-grp 12899 df-mgp 13163 df-ring 13235 df-lmod 13442 df-lssm 13506 df-lsp 13540 |
This theorem is referenced by: (None) |
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