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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 6425 | . . . . . 6 ⊢ + ∈ V |
4 | 3 | tposex 7624 | . . . . 5 ⊢ tpos + ∈ V |
5 | plusgid 16298 | . . . . . 6 ⊢ +g = Slot (+g‘ndx) | |
6 | 5 | setsid 16239 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos + ∈ V) → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
7 | 4, 6 | mpan2 683 | . . . 4 ⊢ (𝑅 ∈ V → tpos + = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉))) |
8 | oppgval.3 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
9 | 2, 8 | oppgval 18089 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
10 | 9 | fveq2i 6414 | . . . 4 ⊢ (+g‘𝑂) = (+g‘(𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
11 | 7, 10 | syl6reqr 2852 | . . 3 ⊢ (𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
12 | tpos0 7620 | . . . . 5 ⊢ tpos ∅ = ∅ | |
13 | 5 | str0 16236 | . . . . 5 ⊢ ∅ = (+g‘∅) |
14 | 12, 13 | eqtr2i 2822 | . . . 4 ⊢ (+g‘∅) = tpos ∅ |
15 | reldmsets 16212 | . . . . . . 7 ⊢ Rel dom sSet | |
16 | 15 | ovprc1 6916 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
17 | 9, 16 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6415 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = (+g‘∅)) |
19 | fvprc 6404 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑅) = ∅) | |
20 | 2, 19 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → + = ∅) |
21 | 20 | tposeqd 7593 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos + = tpos ∅) |
22 | 14, 18, 21 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘𝑂) = tpos + ) |
23 | 11, 22 | pm2.61i 177 | . 2 ⊢ (+g‘𝑂) = tpos + |
24 | 1, 23 | eqtri 2821 | 1 ⊢ ✚ = tpos + |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ∅c0 4115 〈cop 4374 ‘cfv 6101 (class class class)co 6878 tpos ctpos 7589 ndxcnx 16181 sSet csts 16182 +gcplusg 16267 oppgcoppg 18087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addcl 10284 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-tpos 7590 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-nn 11313 df-2 11376 df-ndx 16187 df-slot 16188 df-sets 16191 df-plusg 16280 df-oppg 18088 |
This theorem is referenced by: oppgplus 18091 |
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