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Mirrors > Home > MPE Home > Th. List > frmdval | Structured version Visualization version GIF version |
Description: Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdval.m | ⊢ 𝑀 = (freeMnd‘𝐼) |
frmdval.b | ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) |
frmdval.p | ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
frmdval | ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdval.m | . 2 ⊢ 𝑀 = (freeMnd‘𝐼) | |
2 | df-frmd 18008 | . . 3 ⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) | |
3 | wrdeq 13880 | . . . . . 6 ⊢ (𝑖 = 𝐼 → Word 𝑖 = Word 𝐼) | |
4 | frmdval.b | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) | |
5 | 4 | eqcomd 2827 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 = 𝐵) |
6 | 3, 5 | sylan9eqr 2878 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → Word 𝑖 = 𝐵) |
7 | 6 | opeq2d 4803 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(Base‘ndx), Word 𝑖〉 = 〈(Base‘ndx), 𝐵〉) |
8 | 6 | sqxpeqd 5581 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → (Word 𝑖 × Word 𝑖) = (𝐵 × 𝐵)) |
9 | 8 | reseq2d 5847 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = ( ++ ↾ (𝐵 × 𝐵))) |
10 | frmdval.p | . . . . . 6 ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) | |
11 | 9, 10 | syl6eqr 2874 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = + ) |
12 | 11 | opeq2d 4803 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉 = 〈(+g‘ndx), + 〉) |
13 | 7, 12 | preq12d 4670 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
14 | elex 3512 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
15 | prex 5324 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ∈ V | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ∈ V) |
17 | 2, 13, 14, 16 | fvmptd2 6770 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘𝐼) = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
18 | 1, 17 | syl5eq 2868 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {cpr 4562 〈cop 4566 × cxp 5547 ↾ cres 5551 ‘cfv 6349 Word cword 13855 ++ cconcat 13916 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 freeMndcfrmd 18006 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-frmd 18008 |
This theorem is referenced by: frmdbas 18011 frmdplusg 18013 |
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