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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellspds | Structured version Visualization version GIF version |
Description: Variation on ellspd 20491. (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 6627 | . . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉 | |
8 | f1of 6590 | . . . . 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 6502 | . . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵) |
12 | ellspds.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | 2 | fvexi 6659 | . . . . 5 ⊢ 𝐵 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | 14, 10 | ssexd 5192 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 20491 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))))) |
17 | ssid 3937 | . . . . 5 ⊢ 𝑉 ⊆ 𝑉 | |
18 | resiima 5911 | . . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) | |
19 | 17, 18 | mp1i 13 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) |
20 | 19 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉)) |
21 | 20 | eleq2d 2875 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉))) |
22 | elmapfn 8412 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑m 𝑉) → 𝑎 Fn 𝑉) | |
23 | 22 | adantl 485 | . . . . . . 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 6488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉) |
26 | 15 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → 𝑉 ∈ V) |
27 | inidm 4145 | . . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
28 | eqidd 2799 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣)) | |
29 | fvresi 6912 | . . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣) | |
30 | 29 | adantl 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣) |
31 | 23, 25, 26, 26, 27, 28, 30 | offval 7396 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑎 ∘f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) |
32 | 31 | oveq2d 7151 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))) |
33 | 32 | eqeq2d 2809 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) ↔ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))) |
34 | 33 | anbi2d 631 | . . 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 3255 | . 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 312 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 “ cima 5522 Fn wfn 6319 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 ↑m cmap 8389 finSupp cfsupp 8817 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Σg cgsu 16706 LModclmod 19627 LSpanclspn 19736 |
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-int 4839 df-iun 4883 df-iin 4884 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-se 5479 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-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 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-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lmhm 19787 df-lbs 19840 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-dsmm 20421 df-frlm 20436 df-uvc 20472 |
This theorem is referenced by: elrsp 30989 lbslsp 30992 lbsdiflsp0 31110 fedgmul 31115 |
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