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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 6670 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
4 | mgpval.2 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
6 | 5 | opeq2d 4810 | . . . . 5 ⊢ (𝑟 = 𝑅 → 〈(+g‘ndx), (.r‘𝑟)〉 = 〈(+g‘ndx), · 〉) |
7 | 2, 6 | oveq12d 7174 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
8 | df-mgp 19240 | . . . 4 ⊢ mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉)) | |
9 | ovex 7189 | . . . 4 ⊢ (𝑅 sSet 〈(+g‘ndx), · 〉) ∈ V | |
10 | 7, 8, 9 | fvmpt 6768 | . . 3 ⊢ (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
11 | fvprc 6663 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
12 | reldmsets 16511 | . . . . 5 ⊢ Rel dom sSet | |
13 | 12 | ovprc1 7195 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), · 〉) = ∅) |
14 | 11, 13 | eqtr4d 2859 | . . 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 2844 | 1 ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 〈cop 4573 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 sSet csts 16481 +gcplusg 16565 .rcmulr 16566 mulGrpcmgp 19239 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-sets 16490 df-mgp 19240 |
This theorem is referenced by: mgpplusg 19243 mgplem 19244 mgpress 19250 |
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