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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 18467 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
5 | 4 | fveq2i 6648 | . . . 4 ⊢ (+g‘𝑂) = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
6 | 2 | fvexi 6659 | . . . . . 6 ⊢ + ∈ V |
7 | 6 | tposex 7909 | . . . . 5 ⊢ tpos + ∈ V |
8 | plusgid 16588 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
9 | 8 | setsid 16530 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos + ∈ V) → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
10 | 7, 9 | mpan2 690 | . . . 4 ⊢ (𝑅 ∈ V → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
11 | 5, 10 | eqtr4id 2852 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
12 | tpos0 7905 | . . . . 5 ⊢ tpos ∅ = ∅ | |
13 | 8 | str0 16527 | . . . . 5 ⊢ ∅ = (+g‘∅) |
14 | 12, 13 | eqtr2i 2822 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
15 | reldmsets 16503 | . . . . . . 7 ⊢ Rel dom sSet | |
16 | 15 | ovprc1 7174 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
17 | 4, 16 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = (+g‘∅)) |
19 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑅) = ∅) | |
20 | 2, 19 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
21 | 20 | tposeqd 7878 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
22 | 14, 18, 21 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
23 | 11, 22 | pm2.61i 185 | . 2 ⊢ (+g‘𝑂) = tpos + |
24 | 1, 23 | eqtri 2821 | 1 ⊢ ✚ = tpos + |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 tpos ctpos 7874 ndxcnx 16472 sSet csts 16473 +gcplusg 16557 oppgcoppg 18465 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-sets 16482 df-plusg 16570 df-oppg 18466 |
This theorem is referenced by: oppgplus 18469 |
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