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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 3455 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | xpeq1 5457 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o)) | |
4 | 3 | fveq2d 6542 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o))) |
5 | frgpval.b | . . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) | |
6 | 4, 5 | syl6eqr 2849 | . . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀) |
7 | fveq2 6538 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
8 | frgpval.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
9 | 7, 8 | syl6eqr 2849 | . . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
10 | 6, 9 | oveq12d 7034 | . . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ )) |
11 | df-frgp 18563 | . . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | |
12 | ovex 7048 | . . . 4 ⊢ (𝑀 /s ∼ ) ∈ V | |
13 | 10, 11, 12 | fvmpt 6635 | . . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
15 | 1, 14 | syl5eq 2843 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 × cxp 5441 ‘cfv 6225 (class class class)co 7016 2oc2o 7947 /s cqus 16607 freeMndcfrmd 17823 ~FG cefg 18559 freeGrpcfrgp 18560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-iota 6189 df-fun 6227 df-fv 6233 df-ov 7019 df-frgp 18563 |
This theorem is referenced by: frgp0 18613 frgpeccl 18614 frgpadd 18616 frgpupf 18626 frgpup1 18628 frgpup3lem 18630 frgpnabllem2 18717 |
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