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Mirrors > Home > MPE Home > Th. List > oppgplusfval | Structured version Visualization version GIF version |
Description: Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
Ref | Expression |
---|---|
oppgval.2 | β’ + = (+gβπ ) |
oppgval.3 | β’ π = (oppgβπ ) |
oppgplusfval.4 | β’ β = (+gβπ) |
Ref | Expression |
---|---|
oppgplusfval | β’ β = tpos + |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgplusfval.4 | . 2 β’ β = (+gβπ) | |
2 | oppgval.2 | . . . . . 6 β’ + = (+gβπ ) | |
3 | oppgval.3 | . . . . . 6 β’ π = (oppgβπ ) | |
4 | 2, 3 | oppgval 19133 | . . . . 5 β’ π = (π sSet β¨(+gβndx), tpos + β©) |
5 | 4 | fveq2i 6849 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(+gβndx), tpos + β©)) |
6 | 2 | fvexi 6860 | . . . . . 6 β’ + β V |
7 | 6 | tposex 8195 | . . . . 5 β’ tpos + β V |
8 | plusgid 17168 | . . . . . 6 β’ +g = Slot (+gβndx) | |
9 | 8 | setsid 17088 | . . . . 5 β’ ((π β V β§ tpos + β V) β tpos + = (+gβ(π sSet β¨(+gβndx), tpos + β©))) |
10 | 7, 9 | mpan2 690 | . . . 4 β’ (π β V β tpos + = (+gβ(π sSet β¨(+gβndx), tpos + β©))) |
11 | 5, 10 | eqtr4id 2792 | . . 3 β’ (π β V β (+gβπ) = tpos + ) |
12 | tpos0 8191 | . . . . 5 β’ tpos β = β | |
13 | 8 | str0 17069 | . . . . 5 β’ β = (+gββ ) |
14 | 12, 13 | eqtr2i 2762 | . . . 4 β’ (+gββ ) = tpos β |
15 | reldmsets 17045 | . . . . . . 7 β’ Rel dom sSet | |
16 | 15 | ovprc1 7400 | . . . . . 6 β’ (Β¬ π β V β (π sSet β¨(+gβndx), tpos + β©) = β ) |
17 | 4, 16 | eqtrid 2785 | . . . . 5 β’ (Β¬ π β V β π = β ) |
18 | 17 | fveq2d 6850 | . . . 4 β’ (Β¬ π β V β (+gβπ) = (+gββ )) |
19 | fvprc 6838 | . . . . . 6 β’ (Β¬ π β V β (+gβπ ) = β ) | |
20 | 2, 19 | eqtrid 2785 | . . . . 5 β’ (Β¬ π β V β + = β ) |
21 | 20 | tposeqd 8164 | . . . 4 β’ (Β¬ π β V β tpos + = tpos β ) |
22 | 14, 18, 21 | 3eqtr4a 2799 | . . 3 β’ (Β¬ π β V β (+gβπ) = tpos + ) |
23 | 11, 22 | pm2.61i 182 | . 2 β’ (+gβπ) = tpos + |
24 | 1, 23 | eqtri 2761 | 1 β’ β = tpos + |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3447 β c0 4286 β¨cop 4596 βcfv 6500 (class class class)co 7361 tpos ctpos 8160 sSet csts 17043 ndxcnx 17073 +gcplusg 17141 oppgcoppg 19131 |
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-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-1cn 11117 ax-addcl 11119 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 df-2 12224 df-sets 17044 df-slot 17062 df-ndx 17074 df-plusg 17154 df-oppg 19132 |
This theorem is referenced by: oppgplus 19135 |
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