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Mirrors > Home > MPE Home > Th. List > frgpval | Structured version Visualization version GIF version |
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
frgpval.m | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgpval.b | ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) |
frgpval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
Ref | Expression |
---|---|
frgpval | ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpval.m | . 2 ⊢ 𝐺 = (freeGrp‘𝐼) | |
2 | elex 3512 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | xpeq1 5569 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o)) | |
4 | 3 | fveq2d 6674 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o))) |
5 | frgpval.b | . . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) | |
6 | 4, 5 | syl6eqr 2874 | . . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀) |
7 | fveq2 6670 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
8 | frgpval.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
9 | 7, 8 | syl6eqr 2874 | . . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
10 | 6, 9 | oveq12d 7174 | . . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ )) |
11 | df-frgp 18836 | . . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | |
12 | ovex 7189 | . . . 4 ⊢ (𝑀 /s ∼ ) ∈ V | |
13 | 10, 11, 12 | fvmpt 6768 | . . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
15 | 1, 14 | syl5eq 2868 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 × cxp 5553 ‘cfv 6355 (class class class)co 7156 2oc2o 8096 /s cqus 16778 freeMndcfrmd 18012 ~FG cefg 18832 freeGrpcfrgp 18833 |
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-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-frgp 18836 |
This theorem is referenced by: frgp0 18886 frgpeccl 18887 frgpadd 18889 frgpupf 18899 frgpup1 18901 frgpup3lem 18903 frgpnabllem2 18994 |
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