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Mirrors > Home > MPE Home > Th. List > frgpeccl | Structured version Visualization version GIF version |
Description: Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
frgp0.m | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgp0.r | ⊢ ∼ = ( ~FG ‘𝐼) |
frgpeccl.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
frgpeccl.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
frgpeccl | ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgp0.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
2 | 1 | fvexi 6921 | . . 3 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8812 | . 2 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ (𝑊 / ∼ )) |
4 | frgpeccl.w | . . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | 4 | efgrcl 19748 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
6 | 5 | simpld 494 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V) |
7 | frgp0.m | . . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼) | |
8 | eqid 2735 | . . . . . 6 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
9 | 7, 8, 1 | frgpval 19791 | . . . . 5 ⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
11 | 5 | simprd 495 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
12 | 2on 8519 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7769 | . . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 6, 12, 13 | sylancl 586 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
15 | eqid 2735 | . . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
16 | 8, 15 | frmdbas 18878 | . . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
18 | 11, 17 | eqtr4d 2778 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
19 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → ∼ ∈ V) |
20 | fvexd 6922 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
21 | 10, 18, 19, 20 | qusbas 17592 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = (Base‘𝐺)) |
22 | frgpeccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
23 | 21, 22 | eqtr4di 2793 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = 𝐵) |
24 | 3, 23 | eleqtrd 2841 | 1 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 I cid 5582 × cxp 5687 Oncon0 6386 ‘cfv 6563 (class class class)co 7431 2oc2o 8499 [cec 8742 / cqs 8743 Word cword 14549 Basecbs 17245 /s cqus 17552 freeMndcfrmd 18873 ~FG cefg 19739 freeGrpcfrgp 19740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-imas 17555 df-qus 17556 df-frmd 18875 df-frgp 19743 |
This theorem is referenced by: frgpinv 19797 frgpmhm 19798 vrgpf 19801 frgpup3lem 19810 |
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