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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncoval | Structured version Visualization version GIF version |
Description: Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.) |
Ref | Expression |
---|---|
ltrnel.l | β’ β€ = (leβπΎ) |
ltrnel.a | β’ π΄ = (AtomsβπΎ) |
ltrnel.h | β’ π» = (LHypβπΎ) |
ltrnel.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrncoval | β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β ((πΉ β πΊ)βπ) = (πΉβ(πΊβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1136 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β (πΎ β HL β§ π β π»)) | |
2 | simp2r 1200 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β πΊ β π) | |
3 | eqid 2731 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | ltrnel.h | . . . . 5 β’ π» = (LHypβπΎ) | |
5 | ltrnel.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
6 | 3, 4, 5 | ltrn1o 38693 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β πΊ:(BaseβπΎ)β1-1-ontoβ(BaseβπΎ)) |
7 | 1, 2, 6 | syl2anc 584 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β πΊ:(BaseβπΎ)β1-1-ontoβ(BaseβπΎ)) |
8 | f1of 6804 | . . 3 β’ (πΊ:(BaseβπΎ)β1-1-ontoβ(BaseβπΎ) β πΊ:(BaseβπΎ)βΆ(BaseβπΎ)) | |
9 | 7, 8 | syl 17 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β πΊ:(BaseβπΎ)βΆ(BaseβπΎ)) |
10 | ltrnel.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
11 | 3, 10 | atbase 37857 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
12 | 11 | 3ad2ant3 1135 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β π β (BaseβπΎ)) |
13 | fvco3 6960 | . 2 β’ ((πΊ:(BaseβπΎ)βΆ(BaseβπΎ) β§ π β (BaseβπΎ)) β ((πΉ β πΊ)βπ) = (πΉβ(πΊβπ))) | |
14 | 9, 12, 13 | syl2anc 584 | 1 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β ((πΉ β πΊ)βπ) = (πΉβ(πΊβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β ccom 5657 βΆwf 6512 β1-1-ontoβwf1o 6515 βcfv 6516 Basecbs 17109 lecple 17169 Atomscatm 37831 HLchlt 37918 LHypclh 38553 LTrncltrn 38670 |
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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-map 8789 df-ats 37835 df-laut 38558 df-ldil 38673 df-ltrn 38674 |
This theorem is referenced by: cdlemg41 39287 trlcoabs 39290 trlcoabs2N 39291 trlcolem 39295 cdlemg44 39302 cdlemi2 39388 cdlemk2 39401 cdlemk4 39403 cdlemk8 39407 dia2dimlem4 39636 dihjatcclem3 39989 |
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