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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 | ⊢ + = (+g‘𝑅) |
oppgval.3 | ⊢ 𝑂 = (oppg‘𝑅) |
oppgplusfval.4 | ⊢ ✚ = (+g‘𝑂) |
Ref | Expression |
---|---|
oppgplusfval | ⊢ ✚ = tpos + |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgplusfval.4 | . 2 ⊢ ✚ = (+g‘𝑂) | |
2 | oppgval.2 | . . . . . 6 ⊢ + = (+g‘𝑅) | |
3 | oppgval.3 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
4 | 2, 3 | oppgval 19377 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
5 | 4 | fveq2i 6909 | . . . 4 ⊢ (+g‘𝑂) = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
6 | 2 | fvexi 6920 | . . . . . 6 ⊢ + ∈ V |
7 | 6 | tposex 8283 | . . . . 5 ⊢ tpos + ∈ V |
8 | plusgid 17324 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
9 | 8 | setsid 17241 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos + ∈ V) → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
10 | 7, 9 | mpan2 691 | . . . 4 ⊢ (𝑅 ∈ V → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
11 | 5, 10 | eqtr4id 2793 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
12 | tpos0 8279 | . . . . 5 ⊢ tpos ∅ = ∅ | |
13 | 8 | str0 17222 | . . . . 5 ⊢ ∅ = (+g‘∅) |
14 | 12, 13 | eqtr2i 2763 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
15 | reldmsets 17198 | . . . . . . 7 ⊢ Rel dom sSet | |
16 | 15 | ovprc1 7469 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
17 | 4, 16 | eqtrid 2786 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = (+g‘∅)) |
19 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑅) = ∅) | |
20 | 2, 19 | eqtrid 2786 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
21 | 20 | tposeqd 8252 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
22 | 14, 18, 21 | 3eqtr4a 2800 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (+g‘𝑂) = tpos + |
24 | 1, 23 | eqtri 2762 | 1 ⊢ ✚ = tpos + |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 〈cop 4636 ‘cfv 6562 (class class class)co 7430 tpos ctpos 8248 sSet csts 17196 ndxcnx 17226 +gcplusg 17297 oppgcoppg 19375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-1cn 11210 ax-addcl 11212 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 df-2 12326 df-sets 17197 df-slot 17215 df-ndx 17227 df-plusg 17310 df-oppg 19376 |
This theorem is referenced by: oppgplus 19379 oppgoppcco 48899 |
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