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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 19343 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
5 | 4 | fveq2i 6906 | . . . 4 ⊢ (+g‘𝑂) = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
6 | 2 | fvexi 6917 | . . . . . 6 ⊢ + ∈ V |
7 | 6 | tposex 8277 | . . . . 5 ⊢ tpos + ∈ V |
8 | plusgid 17295 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
9 | 8 | setsid 17212 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos + ∈ V) → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
10 | 7, 9 | mpan2 689 | . . . 4 ⊢ (𝑅 ∈ V → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
11 | 5, 10 | eqtr4id 2785 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
12 | tpos0 8273 | . . . . 5 ⊢ tpos ∅ = ∅ | |
13 | 8 | str0 17193 | . . . . 5 ⊢ ∅ = (+g‘∅) |
14 | 12, 13 | eqtr2i 2755 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
15 | reldmsets 17169 | . . . . . . 7 ⊢ Rel dom sSet | |
16 | 15 | ovprc1 7465 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
17 | 4, 16 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6907 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = (+g‘∅)) |
19 | fvprc 6895 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑅) = ∅) | |
20 | 2, 19 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
21 | 20 | tposeqd 8246 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
22 | 14, 18, 21 | 3eqtr4a 2792 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (+g‘𝑂) = tpos + |
24 | 1, 23 | eqtri 2754 | 1 ⊢ ✚ = tpos + |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 〈cop 4639 ‘cfv 6556 (class class class)co 7426 tpos ctpos 8242 sSet csts 17167 ndxcnx 17197 +gcplusg 17268 oppgcoppg 19341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-1cn 11218 ax-addcl 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-tpos 8243 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-nn 12267 df-2 12329 df-sets 17168 df-slot 17186 df-ndx 17198 df-plusg 17281 df-oppg 19342 |
This theorem is referenced by: oppgplus 19345 |
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