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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 2774 | . . . 4 β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
4 | 3 | pweqd 4619 | . . 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 20627 | . . . . 5 β’ (π β (LSubSpβπΎ) = (LSubSpβπΏ)) |
12 | 11 | rabeqdv 3447 | . . . 4 β’ (π β {π‘ β (LSubSpβπΎ) β£ π β π‘} = {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
13 | 12 | inteqd 4955 | . . 3 β’ (π β β© {π‘ β (LSubSpβπΎ) β£ π β π‘} = β© {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
14 | 4, 13 | mpteq12dv 5239 | . 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 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
17 | eqid 2732 | . . . 4 β’ (LSubSpβπΎ) = (LSubSpβπΎ) | |
18 | eqid 2732 | . . . 4 β’ (LSpanβπΎ) = (LSpanβπΎ) | |
19 | 16, 17, 18 | lspfval 20583 | . . 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 2732 | . . . 4 β’ (BaseβπΏ) = (BaseβπΏ) | |
23 | eqid 2732 | . . . 4 β’ (LSubSpβπΏ) = (LSubSpβπΏ) | |
24 | eqid 2732 | . . . 4 β’ (LSpanβπΏ) = (LSpanβπΏ) | |
25 | 22, 23, 24 | lspfval 20583 | . . 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 2782 | 1 β’ (π β (LSpanβπΎ) = (LSpanβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3432 β wss 3948 π« cpw 4602 β© cint 4950 β¦ cmpt 5231 βcfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 Scalarcsca 17199 Β·π cvsca 17200 LSubSpclss 20541 LSpanclspn 20581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-lss 20542 df-lsp 20582 |
This theorem is referenced by: lbspropd 20709 lidlrsppropd 20854 lindfpropd 32493 |
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