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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 2775 | . . . 4 β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
4 | 3 | pweqd 4578 | . . 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 20493 | . . . . 5 β’ (π β (LSubSpβπΎ) = (LSubSpβπΏ)) |
12 | 11 | rabeqdv 3421 | . . . 4 β’ (π β {π‘ β (LSubSpβπΎ) β£ π β π‘} = {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
13 | 12 | inteqd 4913 | . . 3 β’ (π β β© {π‘ β (LSubSpβπΎ) β£ π β π‘} = β© {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
14 | 4, 13 | mpteq12dv 5197 | . 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 2733 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
17 | eqid 2733 | . . . 4 β’ (LSubSpβπΎ) = (LSubSpβπΎ) | |
18 | eqid 2733 | . . . 4 β’ (LSpanβπΎ) = (LSpanβπΎ) | |
19 | 16, 17, 18 | lspfval 20449 | . . 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 2733 | . . . 4 β’ (BaseβπΏ) = (BaseβπΏ) | |
23 | eqid 2733 | . . . 4 β’ (LSubSpβπΏ) = (LSubSpβπΏ) | |
24 | eqid 2733 | . . . 4 β’ (LSpanβπΏ) = (LSpanβπΏ) | |
25 | 22, 23, 24 | lspfval 20449 | . . 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 2783 | 1 β’ (π β (LSpanβπΎ) = (LSpanβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 β wss 3911 π« cpw 4561 β© cint 4908 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 LSubSpclss 20407 LSpanclspn 20447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lbspropd 20575 lidlrsppropd 20716 lindfpropd 32217 |
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