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 20946. (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 6652 | . . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉 | |
8 | f1of 6615 | . . . . 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 6528 | . . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵) |
12 | ellspds.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | 2 | fvexi 6684 | . . . . 5 ⊢ 𝐵 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | 14, 10 | ssexd 5228 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 20946 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))))) |
17 | ssid 3989 | . . . . 5 ⊢ 𝑉 ⊆ 𝑉 | |
18 | resiima 5944 | . . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) | |
19 | 17, 18 | mp1i 13 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) |
20 | 19 | fveq2d 6674 | . . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉)) |
21 | 20 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉))) |
22 | elmapfn 8429 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑m 𝑉) → 𝑎 Fn 𝑉) | |
23 | 22 | adantl 484 | . . . . . . 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 6515 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉) |
26 | 15 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → 𝑉 ∈ V) |
27 | inidm 4195 | . . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
28 | eqidd 2822 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣)) | |
29 | fvresi 6935 | . . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣) | |
30 | 29 | adantl 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣) |
31 | 23, 25, 26, 26, 27, 28, 30 | offval 7416 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑎 ∘f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) |
32 | 31 | oveq2d 7172 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))) |
33 | 32 | eqeq2d 2832 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉))) ↔ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))) |
34 | 33 | anbi2d 630 | . . 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 3296 | . 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 311 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 I cid 5459 ↾ cres 5557 “ cima 5558 Fn wfn 6350 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ↑m cmap 8406 finSupp cfsupp 8833 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 Σg cgsu 16714 LModclmod 19634 LSpanclspn 19743 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lmhm 19794 df-lbs 19847 df-sra 19944 df-rgmod 19945 df-nzr 20031 df-dsmm 20876 df-frlm 20891 df-uvc 20927 |
This theorem is referenced by: lbslsp 30938 lbsdiflsp0 31022 fedgmul 31027 |
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