![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2768 | . . . 4 β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
4 | 3 | pweqd 4614 | . . 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 20865 | . . . . 5 β’ (π β (LSubSpβπΎ) = (LSubSpβπΏ)) |
12 | 11 | rabeqdv 3441 | . . . 4 β’ (π β {π‘ β (LSubSpβπΎ) β£ π β π‘} = {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
13 | 12 | inteqd 4948 | . . 3 β’ (π β β© {π‘ β (LSubSpβπΎ) β£ π β π‘} = β© {π‘ β (LSubSpβπΏ) β£ π β π‘}) |
14 | 4, 13 | mpteq12dv 5232 | . 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 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
17 | eqid 2726 | . . . 4 β’ (LSubSpβπΎ) = (LSubSpβπΎ) | |
18 | eqid 2726 | . . . 4 β’ (LSpanβπΎ) = (LSpanβπΎ) | |
19 | 16, 17, 18 | lspfval 20820 | . . 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 2726 | . . . 4 β’ (BaseβπΏ) = (BaseβπΏ) | |
23 | eqid 2726 | . . . 4 β’ (LSubSpβπΏ) = (LSubSpβπΏ) | |
24 | eqid 2726 | . . . 4 β’ (LSpanβπΏ) = (LSpanβπΏ) | |
25 | 22, 23, 24 | lspfval 20820 | . . 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 2776 | 1 β’ (π β (LSpanβπΎ) = (LSpanβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 β wss 3943 π« cpw 4597 β© cint 4943 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Scalarcsca 17209 Β·π cvsca 17210 LSubSpclss 20778 LSpanclspn 20818 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-lss 20779 df-lsp 20819 |
This theorem is referenced by: lbspropd 20947 lidlrsppropd 21102 lindfpropd 33004 |
Copyright terms: Public domain | W3C validator |