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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 | . . . . . . 7 ⊢ + = (+g‘𝑅) | |
3 | 2 | fvexi 6686 | . . . . . 6 ⊢ + ∈ V |
4 | 3 | tposex 7928 | . . . . 5 ⊢ tpos + ∈ V |
5 | plusgid 16598 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
6 | 5 | setsid 16540 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos + ∈ V) → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
7 | 4, 6 | mpan2 689 | . . . 4 ⊢ (𝑅 ∈ V → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
8 | oppgval.3 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
9 | 2, 8 | oppgval 18477 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
10 | 9 | fveq2i 6675 | . . . 4 ⊢ (+g‘𝑂) = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
11 | 7, 10 | syl6reqr 2877 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
12 | tpos0 7924 | . . . . 5 ⊢ tpos ∅ = ∅ | |
13 | 5 | str0 16537 | . . . . 5 ⊢ ∅ = (+g‘∅) |
14 | 12, 13 | eqtr2i 2847 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
15 | reldmsets 16513 | . . . . . . 7 ⊢ Rel dom sSet | |
16 | 15 | ovprc1 7197 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
17 | 9, 16 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6676 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = (+g‘∅)) |
19 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑅) = ∅) | |
20 | 2, 19 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
21 | 20 | tposeqd 7897 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
22 | 14, 18, 21 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
23 | 11, 22 | pm2.61i 184 | . 2 ⊢ (+g‘𝑂) = tpos + |
24 | 1, 23 | eqtri 2846 | 1 ⊢ ✚ = tpos + |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 ‘cfv 6357 (class class class)co 7158 tpos ctpos 7893 ndxcnx 16482 sSet csts 16483 +gcplusg 16567 oppgcoppg 18475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-sets 16492 df-plusg 16580 df-oppg 18476 |
This theorem is referenced by: oppgplus 18479 |
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