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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsub0 | Structured version Visualization version GIF version |
Description: The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubccat.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
Ref | Expression |
---|---|
mrsub0 | ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4148 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . 4 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6440 | . . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 145 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
5 | eqid 2778 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
6 | eqid 2778 | . . . . 5 ⊢ (mREx‘𝑇) = (mREx‘𝑇) | |
7 | 5, 6, 2 | mrsubff 32008 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇))) |
8 | ffun 6294 | . . . 4 ⊢ (𝑆:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇)) → Fun 𝑆) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆) |
10 | 5, 6, 2 | mrsubrn 32009 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) |
11 | 10 | eleq2i 2851 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇)))) |
12 | 11 | biimpi 208 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇)))) |
13 | fvelima 6508 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇)))) → ∃𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 579 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8162 | . . . . . . 7 ⊢ (𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) | |
16 | 15 | adantl 475 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) |
17 | ssidd 3843 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → (mVR‘𝑇) ⊆ (mVR‘𝑇)) | |
18 | wrd0 13627 | . . . . . . 7 ⊢ ∅ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) | |
19 | eqid 2778 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
20 | 19, 5, 6 | mrexval 31997 | . . . . . . . 8 ⊢ (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
21 | 20 | adantr 474 | . . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
22 | 18, 21 | syl5eleqr 2866 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → ∅ ∈ (mREx‘𝑇)) |
23 | eqid 2778 | . . . . . . 7 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) | |
24 | 19, 5, 6, 2, 23 | mrsubval 32005 | . . . . . 6 ⊢ ((𝑓:(mVR‘𝑇)⟶(mREx‘𝑇) ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ ∅ ∈ (mREx‘𝑇)) → ((𝑆‘𝑓)‘∅) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅))) |
25 | 16, 17, 22, 24 | syl3anc 1439 | . . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → ((𝑆‘𝑓)‘∅) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅))) |
26 | co02 5903 | . . . . . . 7 ⊢ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅) = ∅ | |
27 | 26 | oveq2i 6933 | . . . . . 6 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ∅) |
28 | 23 | frmd0 17784 | . . . . . . 7 ⊢ ∅ = (0g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) |
29 | 28 | gsum0 17664 | . . . . . 6 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ∅) = ∅ |
30 | 27, 29 | eqtri 2802 | . . . . 5 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅)) = ∅ |
31 | 25, 30 | syl6eq 2830 | . . . 4 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → ((𝑆‘𝑓)‘∅) = ∅) |
32 | fveq1 6445 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘∅) = (𝐹‘∅)) | |
33 | 32 | eqeq1d 2780 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘∅) = ∅ ↔ (𝐹‘∅) = ∅)) |
34 | 31, 33 | syl5ibcom 237 | . . 3 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘∅) = ∅)) |
35 | 34 | rexlimdva 3213 | . 2 ⊢ (𝑇 ∈ V → (∃𝑓 ∈ ((mREx‘𝑇) ↑𝑚 (mVR‘𝑇))(𝑆‘𝑓) = 𝐹 → (𝐹‘∅) = ∅)) |
36 | 4, 14, 35 | sylc 65 | 1 ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 Vcvv 3398 ∪ cun 3790 ⊆ wss 3792 ∅c0 4141 ifcif 4307 ↦ cmpt 4965 ran crn 5356 “ cima 5358 ∘ ccom 5359 Fun wfun 6129 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑𝑚 cmap 8140 ↑pm cpm 8141 Word cword 13599 〈“cs1 13685 Σg cgsu 16487 freeMndcfrmd 17771 mCNcmcn 31956 mVRcmvar 31957 mRExcmrex 31962 mRSubstcmrsub 31966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-gsum 16489 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-frmd 17773 df-mrex 31982 df-mrsub 31986 |
This theorem is referenced by: mrsubvrs 32018 |
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