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Mirrors > Home > MPE Home > Th. List > lsppropd | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.) |
Ref | Expression |
---|---|
lsspropd.b1 | β’ (π β π΅ = (BaseβπΎ)) |
lsspropd.b2 | β’ (π β π΅ = (BaseβπΏ)) |
lsspropd.w | β’ (π β π΅ β π) |
lsspropd.p | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
lsspropd.s1 | β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) β π) |
lsspropd.s2 | β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) |
lsspropd.p1 | β’ (π β π = (Baseβ(ScalarβπΎ))) |
lsspropd.p2 | β’ (π β π = (Baseβ(ScalarβπΏ))) |
lsppropd.v1 | β’ (π β πΎ β π) |
lsppropd.v2 | β’ (π β πΏ β π) |
Ref | Expression |
---|---|
lsppropd | β’ (π β (LSpanβπΎ) = (LSpanβπΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsspropd.b1 | . . . . 5 β’ (π β π΅ = (BaseβπΎ)) | |
2 | lsspropd.b2 | . . . . 5 β’ (π β π΅ = (BaseβπΏ)) | |
3 | 1, 2 | eqtr3d 2770 | . . . 4 β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
4 | 3 | pweqd 4623 | . . 3 β’ (π β π« (BaseβπΎ) = π« (BaseβπΏ)) |
5 | lsspropd.w | . . . . . 6 β’ (π β π΅ β π) | |
6 | lsspropd.p | . . . . . 6 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) | |
7 | lsspropd.s1 | . . . . . 6 β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) β π) | |
8 | lsspropd.s2 | . . . . . 6 β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) | |
9 | lsspropd.p1 | . . . . . 6 β’ (π β π = (Baseβ(ScalarβπΎ))) | |
10 | lsspropd.p2 | . . . . . 6 β’ (π β π = (Baseβ(ScalarβπΏ))) | |
11 | 1, 2, 5, 6, 7, 8, 9, 10 | lsspropd 20916 | . . . . 5 β’ (π β (LSubSpβπΎ) = (LSubSpβπΏ)) |
12 | 11 | rabeqdv 3446 | . . . 4 β’ (π β {π‘ β (LSubSpβπΎ) β£ π β π‘} = {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
13 | 12 | inteqd 4958 | . . 3 β’ (π β β© {π‘ β (LSubSpβπΎ) β£ π β π‘} = β© {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
14 | 4, 13 | mpteq12dv 5243 | . 2 β’ (π β (π β π« (BaseβπΎ) β¦ β© {π‘ β (LSubSpβπΎ) β£ π β π‘}) = (π β π« (BaseβπΏ) β¦ β© {π‘ β (LSubSpβπΏ) β£ π β π‘})) |
15 | lsppropd.v1 | . . 3 β’ (π β πΎ β π) | |
16 | eqid 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
17 | eqid 2728 | . . . 4 β’ (LSubSpβπΎ) = (LSubSpβπΎ) | |
18 | eqid 2728 | . . . 4 β’ (LSpanβπΎ) = (LSpanβπΎ) | |
19 | 16, 17, 18 | lspfval 20871 | . . 3 β’ (πΎ β π β (LSpanβπΎ) = (π β π« (BaseβπΎ) β¦ β© {π‘ β (LSubSpβπΎ) β£ π β π‘})) |
20 | 15, 19 | syl 17 | . 2 β’ (π β (LSpanβπΎ) = (π β π« (BaseβπΎ) β¦ β© {π‘ β (LSubSpβπΎ) β£ π β π‘})) |
21 | lsppropd.v2 | . . 3 β’ (π β πΏ β π) | |
22 | eqid 2728 | . . . 4 β’ (BaseβπΏ) = (BaseβπΏ) | |
23 | eqid 2728 | . . . 4 β’ (LSubSpβπΏ) = (LSubSpβπΏ) | |
24 | eqid 2728 | . . . 4 β’ (LSpanβπΏ) = (LSpanβπΏ) | |
25 | 22, 23, 24 | lspfval 20871 | . . 3 β’ (πΏ β π β (LSpanβπΏ) = (π β π« (BaseβπΏ) β¦ β© {π‘ β (LSubSpβπΏ) β£ π β π‘})) |
26 | 21, 25 | syl 17 | . 2 β’ (π β (LSpanβπΏ) = (π β π« (BaseβπΏ) β¦ β© {π‘ β (LSubSpβπΏ) β£ π β π‘})) |
27 | 14, 20, 26 | 3eqtr4d 2778 | 1 β’ (π β (LSpanβπΎ) = (LSpanβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3430 β wss 3949 π« cpw 4606 β© cint 4953 β¦ cmpt 5235 βcfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 Scalarcsca 17245 Β·π cvsca 17246 LSubSpclss 20829 LSpanclspn 20869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-lss 20830 df-lsp 20870 |
This theorem is referenced by: lbspropd 20998 lidlrsppropd 21153 lindfpropd 33130 |
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