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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 19365 | . . . . 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 8285 | . . . . 5 ⊢ tpos + ∈ V |
| 8 | plusgid 17324 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
| 9 | 8 | setsid 17244 | . . . . 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 2796 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
| 12 | tpos0 8281 | . . . . 5 ⊢ tpos ∅ = ∅ | |
| 13 | 8 | str0 17226 | . . . . 5 ⊢ ∅ = (+g‘∅) |
| 14 | 12, 13 | eqtr2i 2766 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
| 15 | reldmsets 17202 | . . . . . . 7 ⊢ Rel dom sSet | |
| 16 | 15 | ovprc1 7470 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
| 17 | 4, 16 | eqtrid 2789 | . . . . 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 2789 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
| 21 | 20 | tposeqd 8254 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
| 22 | 14, 18, 21 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
| 23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (+g‘𝑂) = tpos + |
| 24 | 1, 23 | eqtri 2765 | 1 ⊢ ✚ = tpos + |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 sSet csts 17200 ndxcnx 17230 +gcplusg 17297 oppgcoppg 19363 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-plusg 17310 df-oppg 19364 |
| This theorem is referenced by: oppgplus 19367 oppgoppcco 49188 |
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