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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 3501 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
| 3 | xpeq1 5699 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o)) | |
| 4 | 3 | fveq2d 6910 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o))) |
| 5 | frgpval.b | . . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀) |
| 7 | fveq2 6906 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
| 8 | frgpval.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
| 10 | 6, 9 | oveq12d 7449 | . . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ )) |
| 11 | df-frgp 19728 | . . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | |
| 12 | ovex 7464 | . . . 4 ⊢ (𝑀 /s ∼ ) ∈ V | |
| 13 | 10, 11, 12 | fvmpt 7016 | . . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
| 14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
| 15 | 1, 14 | eqtrid 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 × cxp 5683 ‘cfv 6561 (class class class)co 7431 2oc2o 8500 /s cqus 17550 freeMndcfrmd 18860 ~FG cefg 19724 freeGrpcfrgp 19725 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-frgp 19728 |
| This theorem is referenced by: frgp0 19778 frgpeccl 19779 frgpadd 19781 frgpupf 19791 frgpup1 19793 frgpup3lem 19795 frgpnabllem2 19892 |
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