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Mirrors > Home > MPE Home > Th. List > mgpval | Structured version Visualization version GIF version |
Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
mgpval.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgpval.2 | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
mgpval | ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), · 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpval.1 | . 2 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
3 | fveq2 6672 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
4 | mgpval.2 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2876 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
6 | 5 | opeq2d 4812 | . . . . 5 ⊢ (𝑟 = 𝑅 → 〈(+g‘ndx), (.r‘𝑟)〉 = 〈(+g‘ndx), · 〉) |
7 | 2, 6 | oveq12d 7176 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
8 | df-mgp 19242 | . . . 4 ⊢ mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉)) | |
9 | ovex 7191 | . . . 4 ⊢ (𝑅 sSet 〈(+g‘ndx), · 〉) ∈ V | |
10 | 7, 8, 9 | fvmpt 6770 | . . 3 ⊢ (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
11 | fvprc 6665 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
12 | reldmsets 16513 | . . . . 5 ⊢ Rel dom sSet | |
13 | 12 | ovprc1 7197 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), · 〉) = ∅) |
14 | 11, 13 | eqtr4d 2861 | . . 3 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
15 | 10, 14 | pm2.61i 184 | . 2 ⊢ (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉) |
16 | 1, 15 | eqtri 2846 | 1 ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), · 〉) |
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 ndxcnx 16482 sSet csts 16483 +gcplusg 16567 .rcmulr 16568 mulGrpcmgp 19241 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-sets 16492 df-mgp 19242 |
This theorem is referenced by: mgpplusg 19245 mgplem 19246 mgpress 19252 |
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