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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 18230 | . . 3 ⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) | |
3 | wrdeq 14056 | . . . . . 6 ⊢ (𝑖 = 𝐼 → Word 𝑖 = Word 𝐼) | |
4 | frmdval.b | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) | |
5 | 4 | eqcomd 2742 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 = 𝐵) |
6 | 3, 5 | sylan9eqr 2793 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → Word 𝑖 = 𝐵) |
7 | 6 | opeq2d 4777 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(Base‘ndx), Word 𝑖〉 = 〈(Base‘ndx), 𝐵〉) |
8 | 6 | sqxpeqd 5568 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → (Word 𝑖 × Word 𝑖) = (𝐵 × 𝐵)) |
9 | 8 | reseq2d 5836 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = ( ++ ↾ (𝐵 × 𝐵))) |
10 | frmdval.p | . . . . . 6 ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) | |
11 | 9, 10 | eqtr4di 2789 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = + ) |
12 | 11 | opeq2d 4777 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉 = 〈(+g‘ndx), + 〉) |
13 | 7, 12 | preq12d 4643 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
14 | elex 3416 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
15 | prex 5310 | . . . 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 6804 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘𝐼) = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
18 | 1, 17 | syl5eq 2783 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {cpr 4529 〈cop 4533 × cxp 5534 ↾ cres 5538 ‘cfv 6358 Word cword 14034 ++ cconcat 14090 ndxcnx 16663 Basecbs 16666 +gcplusg 16749 freeMndcfrmd 18228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-frmd 18230 |
This theorem is referenced by: frmdbas 18233 frmdplusg 18235 |
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