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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 6191 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
4 | mgpval.2 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
6 | 5 | opeq2d 4409 | . . . . 5 ⊢ (𝑟 = 𝑅 → 〈(+g‘ndx), (.r‘𝑟)〉 = 〈(+g‘ndx), · 〉) |
7 | 2, 6 | oveq12d 6668 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
8 | df-mgp 18490 | . . . 4 ⊢ mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet 〈(+g‘ndx), (.r‘𝑟)〉)) | |
9 | ovex 6678 | . . . 4 ⊢ (𝑅 sSet 〈(+g‘ndx), · 〉) ∈ V | |
10 | 7, 8, 9 | fvmpt 6282 | . . 3 ⊢ (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
11 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
12 | reldmsets 15886 | . . . . 5 ⊢ Rel dom sSet | |
13 | 12 | ovprc1 6684 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), · 〉) = ∅) |
14 | 11, 13 | eqtr4d 2659 | . . 3 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉)) |
15 | 10, 14 | pm2.61i 176 | . 2 ⊢ (mulGrp‘𝑅) = (𝑅 sSet 〈(+g‘ndx), · 〉) |
16 | 1, 15 | eqtri 2644 | 1 ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 +gcplusg 15941 .rcmulr 15942 mulGrpcmgp 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-sets 15864 df-mgp 18490 |
This theorem is referenced by: mgpplusg 18493 mgplem 18494 mgpress 18500 |
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