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Mirrors > Home > ILE Home > Th. List > ressipg | GIF version |
Description: The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
resssca.1 | β’ π» = (πΊ βΎs π΄) |
ressip.2 | β’ , = (Β·πβπΊ) |
Ref | Expression |
---|---|
ressipg | β’ ((πΊ β π β§ π΄ β π) β , = (Β·πβπ»)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resssca.1 | . 2 β’ π» = (πΊ βΎs π΄) | |
2 | ressip.2 | . 2 β’ , = (Β·πβπΊ) | |
3 | ipslid 12643 | . 2 β’ (Β·π = Slot (Β·πβndx) β§ (Β·πβndx) β β) | |
4 | ipndxnbasendx 12644 | . 2 β’ (Β·πβndx) β (Baseβndx) | |
5 | simpl 109 | . 2 β’ ((πΊ β π β§ π΄ β π) β πΊ β π) | |
6 | simpr 110 | . 2 β’ ((πΊ β π β§ π΄ β π) β π΄ β π) | |
7 | 1, 2, 3, 4, 5, 6 | resseqnbasd 12546 | 1 β’ ((πΊ β π β§ π΄ β π) β , = (Β·πβπ»)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2158 βcfv 5228 (class class class)co 5888 βΎs cress 12476 Β·πcip 12555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-iress 12483 df-ip 12568 |
This theorem is referenced by: (None) |
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