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Mirrors > Home > MPE Home > Th. List > frmdbas | Structured version Visualization version GIF version |
Description: The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
frmdbas.m | ⊢ 𝑀 = (freeMnd‘𝐼) |
frmdbas.b | ⊢ 𝐵 = (Base‘𝑀) |
Ref | Expression |
---|---|
frmdbas | ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdbas.b | . 2 ⊢ 𝐵 = (Base‘𝑀) | |
2 | frmdbas.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
3 | eqidd 2727 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 = Word 𝐼) | |
4 | eqid 2726 | . . . . 5 ⊢ ( ++ ↾ (Word 𝐼 × Word 𝐼)) = ( ++ ↾ (Word 𝐼 × Word 𝐼)) | |
5 | 2, 3, 4 | frmdval 18773 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩}) |
6 | 5 | fveq2d 6888 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘{⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩})) |
7 | wrdexg 14477 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 ∈ V) | |
8 | eqid 2726 | . . . . 5 ⊢ {⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩} = {⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩} | |
9 | 8 | grpbase 17237 | . . . 4 ⊢ (Word 𝐼 ∈ V → Word 𝐼 = (Base‘{⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩})) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 = (Base‘{⟨(Base‘ndx), Word 𝐼⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝐼 × Word 𝐼))⟩})) |
11 | 6, 10 | eqtr4d 2769 | . 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼) |
12 | 1, 11 | eqtrid 2778 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {cpr 4625 ⟨cop 4629 × cxp 5667 ↾ cres 5671 ‘cfv 6536 Word cword 14467 ++ cconcat 14523 ndxcnx 17132 Basecbs 17150 +gcplusg 17203 freeMndcfrmd 18769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-frmd 18771 |
This theorem is referenced by: frmdelbas 18775 frmdplusg 18776 frmdmnd 18781 frmd0 18782 frmdsssubm 18783 frmdgsum 18784 frmdup1 18786 frmdup3lem 18788 frmdup3 18789 frgpcpbl 19676 frgp0 19677 frgpeccl 19678 frgpadd 19680 frgpmhm 19682 frgpupf 19690 frgpup1 19692 frgpup3lem 19694 frgpnabllem2 19791 mrsubcv 35028 mrsubff 35030 mrsubccat 35036 elmrsubrn 35038 |
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