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Mirrors > Home > MPE Home > Th. List > opprlem | Structured version Visualization version GIF version |
Description: Lemma for opprbas 20064 and oppradd 20066. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) |
Ref | Expression |
---|---|
opprbas.1 | β’ π = (opprβπ ) |
opprlem.2 | β’ πΈ = Slot (πΈβndx) |
opprlem.3 | β’ (πΈβndx) β (.rβndx) |
Ref | Expression |
---|---|
opprlem | β’ (πΈβπ ) = (πΈβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprlem.2 | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | opprlem.3 | . . 3 β’ (πΈβndx) β (.rβndx) | |
3 | 1, 2 | setsnid 17089 | . 2 β’ (πΈβπ ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
4 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
5 | eqid 2733 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
6 | opprbas.1 | . . . 4 β’ π = (opprβπ ) | |
7 | 4, 5, 6 | opprval 20058 | . . 3 β’ π = (π sSet β¨(.rβndx), tpos (.rβπ )β©) |
8 | 7 | fveq2i 6849 | . 2 β’ (πΈβπ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
9 | 3, 8 | eqtr4i 2764 | 1 β’ (πΈβπ ) = (πΈβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wne 2940 β¨cop 4596 βcfv 6500 (class class class)co 7361 tpos ctpos 8160 sSet csts 17043 Slot cslot 17061 ndxcnx 17073 Basecbs 17091 .rcmulr 17142 opprcoppr 20056 |
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-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-res 5649 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-tpos 8161 df-sets 17044 df-slot 17062 df-oppr 20057 |
This theorem is referenced by: opprbas 20064 oppradd 20066 |
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