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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 18024 | . . . 4 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝑌 ∈ 𝐼) → (𝑈‘𝑌) = 〈“𝑌”〉) |
5 | 1, 2, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑈‘𝑌) = 〈“𝑌”〉) |
6 | 5 | fveq2d 6676 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐸‘〈“𝑌”〉)) |
7 | 2 | s1cld 13959 | . . . 4 ⊢ (𝜑 → 〈“𝑌”〉 ∈ Word 𝐼) |
8 | coeq2 5731 | . . . . . 6 ⊢ (𝑥 = 〈“𝑌”〉 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 〈“𝑌”〉)) | |
9 | 8 | oveq2d 7174 | . . . . 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 7191 | . . . . 5 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V | |
12 | 9, 10, 11 | fvmpt3i 6775 | . . . 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 14197 | . . . . 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 7174 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐴 ∘ 〈“𝑌”〉)) = (𝐺 Σg 〈“(𝐴‘𝑌)”〉)) |
18 | 14, 2 | ffvelrnd 6854 | . . . 4 ⊢ (𝜑 → (𝐴‘𝑌) ∈ 𝐵) |
19 | frmdup.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
20 | 19 | gsumws1 18004 | . . . 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 2862 | . 2 ⊢ (𝜑 → (𝐸‘〈“𝑌”〉) = (𝐴‘𝑌)) |
23 | 6, 22 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Word cword 13864 〈“cs1 13951 Basecbs 16485 Σg cgsu 16716 Mndcmnd 17913 freeMndcfrmd 18014 varFMndcvrmd 18015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-word 13865 df-s1 13952 df-0g 16717 df-gsum 16718 df-vrmd 18017 |
This theorem is referenced by: frmdup3 18034 |
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