![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ellspds | Structured version Visualization version GIF version |
Description: Variation on ellspd 21845. (Contributed by Thierry Arnoux, 18-May-2023.) |
Ref | Expression |
---|---|
ellspds.n | ⊢ 𝑁 = (LSpan‘𝑀) |
ellspds.v | ⊢ 𝐵 = (Base‘𝑀) |
ellspds.k | ⊢ 𝐾 = (Base‘𝑆) |
ellspds.s | ⊢ 𝑆 = (Scalar‘𝑀) |
ellspds.z | ⊢ 0 = (0g‘𝑆) |
ellspds.t | ⊢ · = ( ·𝑠 ‘𝑀) |
ellspds.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
ellspds.1 | ⊢ (𝜑 → 𝑉 ⊆ 𝐵) |
Ref | Expression |
---|---|
ellspds | ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspds.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑀) | |
2 | ellspds.v | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
3 | ellspds.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
4 | ellspds.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑀) | |
5 | ellspds.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
6 | ellspds.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑀) | |
7 | f1oi 6900 | . . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉 | |
8 | f1of 6862 | . . . . 5 ⊢ (( I ↾ 𝑉):𝑉–1-1-onto→𝑉 → ( I ↾ 𝑉):𝑉⟶𝑉) | |
9 | 7, 8 | mp1i 13 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝑉) |
10 | ellspds.1 | . . . 4 ⊢ (𝜑 → 𝑉 ⊆ 𝐵) | |
11 | 9, 10 | fssd 6764 | . . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵) |
12 | ellspds.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | 2 | fvexi 6934 | . . . . 5 ⊢ 𝐵 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | 14, 10 | ssexd 5342 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 21845 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))))) |
17 | ssid 4031 | . . . . 5 ⊢ 𝑉 ⊆ 𝑉 | |
18 | resiima 6105 | . . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) | |
19 | 17, 18 | mp1i 13 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) |
20 | 19 | fveq2d 6924 | . . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉)) |
21 | 20 | eleq2d 2830 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉))) |
22 | elmapfn 8923 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑m 𝑉) → 𝑎 Fn 𝑉) | |
23 | 22 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → 𝑎 Fn 𝑉) |
24 | 7, 8 | mp1i 13 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ( I ↾ 𝑉):𝑉⟶𝑉) |
25 | 24 | ffnd 6748 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉) |
26 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → 𝑉 ∈ V) |
27 | inidm 4248 | . . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
28 | eqidd 2741 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣)) | |
29 | fvresi 7207 | . . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣) | |
30 | 29 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣) |
31 | 23, 25, 26, 26, 27, 28, 30 | offval 7723 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑎 ∘f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) |
32 | 31 | oveq2d 7464 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))) |
33 | 32 | eqeq2d 2751 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) ↔ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))) |
34 | 33 | anbi2d 629 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ((𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))) ↔ (𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
35 | 34 | rexbidva 3183 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
36 | 16, 21, 35 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 I cid 5592 ↾ cres 5702 “ cima 5703 Fn wfn 6568 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 ∘f cof 7712 ↑m cmap 8884 finSupp cfsupp 9431 Basecbs 17258 Scalarcsca 17314 ·𝑠 cvsca 17315 0gc0g 17499 Σg cgsu 17500 LModclmod 20880 LSpanclspn 20992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-nzr 20539 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lmhm 21044 df-lbs 21097 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-uvc 21826 |
This theorem is referenced by: elrsp 33365 lbslsp 33370 lbsdiflsp0 33639 fedgmul 33644 |
Copyright terms: Public domain | W3C validator |