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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 17748 | . . . 4 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝑌 ∈ 𝐼) → (𝑈‘𝑌) = 〈“𝑌”〉) |
5 | 1, 2, 4 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑈‘𝑌) = 〈“𝑌”〉) |
6 | 5 | fveq2d 6437 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐸‘〈“𝑌”〉)) |
7 | 2 | s1cld 13663 | . . . 4 ⊢ (𝜑 → 〈“𝑌”〉 ∈ Word 𝐼) |
8 | coeq2 5513 | . . . . . 6 ⊢ (𝑥 = 〈“𝑌”〉 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 〈“𝑌”〉)) | |
9 | 8 | oveq2d 6921 | . . . . 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 6937 | . . . . 5 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V | |
12 | 9, 10, 11 | fvmpt3i 6534 | . . . 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 13954 | . . . . 5 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) | |
16 | 2, 14, 15 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝐴 ∘ 〈“𝑌”〉) = 〈“(𝐴‘𝑌)”〉) |
17 | 16 | oveq2d 6921 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉)) = (𝐺 Σg 〈“(𝐴‘𝑌)”〉)) |
18 | 14, 2 | ffvelrnd 6609 | . . . 4 ⊢ (𝜑 → (𝐴‘𝑌) ∈ 𝐵) |
19 | frmdup.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
20 | 19 | gsumws1 17729 | . . . 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 2865 | . 2 ⊢ (𝜑 → (𝐸‘〈“𝑌”〉) = (𝐴‘𝑌)) |
23 | 6, 22 | eqtrd 2861 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ↦ cmpt 4952 ∘ ccom 5346 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 Word cword 13574 〈“cs1 13655 Basecbs 16222 Σg cgsu 16454 Mndcmnd 17647 freeMndcfrmd 17738 varFMndcvrmd 17739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-word 13575 df-s1 13656 df-0g 16455 df-gsum 16456 df-vrmd 17741 |
This theorem is referenced by: frmdup3 17758 |
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