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Mirrors > Home > MPE Home > Th. List > frmdup2 | Structured version Visualization version GIF version |
Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdup.m | ⊢ 𝑀 = (freeMnd‘𝐼) |
frmdup.b | ⊢ 𝐵 = (Base‘𝐺) |
frmdup.e | ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) |
frmdup.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
frmdup.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
frmdup.a | ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) |
frmdup2.u | ⊢ 𝑈 = (varFMnd‘𝐼) |
frmdup2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐼) |
Ref | Expression |
---|---|
frmdup2 | ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdup.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
2 | frmdup2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
3 | frmdup2.u | . . . . 5 ⊢ 𝑈 = (varFMnd‘𝐼) | |
4 | 3 | vrmdval 18021 | . . . 4 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝑌 ∈ 𝐼) → (𝑈‘𝑌) = 〈“𝑌”〉) |
5 | 1, 2, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑈‘𝑌) = 〈“𝑌”〉) |
6 | 5 | fveq2d 6673 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐸‘〈“𝑌”〉)) |
7 | 2 | s1cld 13956 | . . . 4 ⊢ (𝜑 → 〈“𝑌”〉 ∈ Word 𝐼) |
8 | coeq2 5728 | . . . . . 6 ⊢ (𝑥 = 〈“𝑌”〉 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 〈“𝑌”〉)) | |
9 | 8 | oveq2d 7171 | . . . . 5 ⊢ (𝑥 = 〈“𝑌”〉 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉))) |
10 | frmdup.e | . . . . 5 ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) | |
11 | ovex 7188 | . . . . 5 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V | |
12 | 9, 10, 11 | fvmpt3i 6772 | . . . 4 ⊢ (〈“𝑌”〉 ∈ Word 𝐼 → (𝐸‘〈“𝑌”〉) = (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉))) |
13 | 7, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸‘〈“𝑌”〉) = (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉))) |
14 | frmdup.a | . . . . 5 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) | |
15 | s1co 14194 | . . . . 5 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) | |
16 | 2, 14, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) |
17 | 16 | oveq2d 7171 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉)) = (𝐺 Σg 〈“(𝐴‘𝑌)”〉)) |
18 | 14, 2 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝐴‘𝑌) ∈ 𝐵) |
19 | frmdup.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
20 | 19 | gsumws1 18001 | . . . 4 ⊢ ((𝐴‘𝑌) ∈ 𝐵 → (𝐺 Σg 〈“(𝐴‘𝑌)”〉) = (𝐴‘𝑌)) |
21 | 18, 20 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 Σg 〈“(𝐴‘𝑌)”〉) = (𝐴‘𝑌)) |
22 | 13, 17, 21 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → (𝐸‘〈“𝑌”〉) = (𝐴‘𝑌)) |
23 | 6, 22 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5145 ∘ ccom 5558 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 Word cword 13860 〈“cs1 13948 Basecbs 16482 Σg cgsu 16713 Mndcmnd 17910 freeMndcfrmd 18011 varFMndcvrmd 18012 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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-rmo 3146 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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-word 13861 df-s1 13949 df-0g 16714 df-gsum 16715 df-vrmd 18014 |
This theorem is referenced by: frmdup3 18031 |
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