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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 18014 | . . . 4 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝑌 ∈ 𝐼) → (𝑈‘𝑌) = 〈“𝑌”〉) |
5 | 1, 2, 4 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑈‘𝑌) = 〈“𝑌”〉) |
6 | 5 | fveq2d 6649 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐸‘〈“𝑌”〉)) |
7 | 2 | s1cld 13948 | . . . 4 ⊢ (𝜑 → 〈“𝑌”〉 ∈ Word 𝐼) |
8 | coeq2 5693 | . . . . . 6 ⊢ (𝑥 = 〈“𝑌”〉 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 〈“𝑌”〉)) | |
9 | 8 | oveq2d 7151 | . . . . 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 7168 | . . . . 5 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V | |
12 | 9, 10, 11 | fvmpt3i 6750 | . . . 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 14186 | . . . . 5 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) | |
16 | 2, 14, 15 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) |
17 | 16 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉)) = (𝐺 Σg 〈“(𝐴‘𝑌)”〉)) |
18 | 14, 2 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝐴‘𝑌) ∈ 𝐵) |
19 | frmdup.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
20 | 19 | gsumws1 17994 | . . . 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 2837 | . 2 ⊢ (𝜑 → (𝐸‘〈“𝑌”〉) = (𝐴‘𝑌)) |
23 | 6, 22 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Word cword 13857 〈“cs1 13940 Basecbs 16475 Σg cgsu 16706 Mndcmnd 17903 freeMndcfrmd 18004 varFMndcvrmd 18005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-word 13858 df-s1 13941 df-0g 16707 df-gsum 16708 df-vrmd 18007 |
This theorem is referenced by: frmdup3 18024 |
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