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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 17209 | . . . 4 ⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝑉 → freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉})) |
4 | wrdeq 13182 | . . . . . 6 ⊢ (𝑖 = 𝐼 → Word 𝑖 = Word 𝐼) | |
5 | frmdval.b | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) | |
6 | 5 | eqcomd 2616 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → Word 𝐼 = 𝐵) |
7 | 4, 6 | sylan9eqr 2666 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → Word 𝑖 = 𝐵) |
8 | 7 | opeq2d 4347 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(Base‘ndx), Word 𝑖〉 = 〈(Base‘ndx), 𝐵〉) |
9 | 7 | sqxpeqd 5065 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → (Word 𝑖 × Word 𝑖) = (𝐵 × 𝐵)) |
10 | 9 | reseq2d 5317 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = ( ++ ↾ (𝐵 × 𝐵))) |
11 | frmdval.p | . . . . . 6 ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) | |
12 | 10, 11 | syl6eqr 2662 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → ( ++ ↾ (Word 𝑖 × Word 𝑖)) = + ) |
13 | 12 | opeq2d 4347 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉 = 〈(+g‘ndx), + 〉) |
14 | 8, 13 | preq12d 4220 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼) → {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
15 | elex 3185 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
16 | prex 4836 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ∈ V | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ∈ V) |
18 | 3, 14, 15, 17 | fvmptd 6197 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘𝐼) = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
19 | 1, 18 | syl5eq 2656 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {cpr 4127 〈cop 4131 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ‘cfv 5804 Word cword 13146 ++ cconcat 13148 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 freeMndcfrmd 17207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-frmd 17209 |
This theorem is referenced by: frmdbas 17212 frmdplusg 17214 |
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