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 20940. (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 6646 | . . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉 | |
8 | f1of 6609 | . . . . 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 6522 | . . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵) |
12 | ellspds.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | 2 | fvexi 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | 14, 10 | ssexd 5220 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 20940 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑎 ∘f · ( I ↾ 𝑉)))))) |
17 | ssid 3988 | . . . . 5 ⊢ 𝑉 ⊆ 𝑉 | |
18 | resiima 5938 | . . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) | |
19 | 17, 18 | mp1i 13 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉) |
20 | 19 | fveq2d 6668 | . . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉)) |
21 | 20 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉))) |
22 | elmapfn 8423 | . . . . . . . 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 6509 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉) |
26 | 15 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → 𝑉 ∈ V) |
27 | inidm 4194 | . . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
28 | eqidd 2822 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣)) | |
29 | fvresi 6929 | . . . . . . . 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 7410 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑m 𝑉)) → (𝑎 ∘f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) |
32 | 31 | oveq2d 7166 | . . . . 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 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 ↾ cres 5551 “ cima 5552 Fn wfn 6344 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 ↑m cmap 8400 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Σg cgsu 16708 LModclmod 19628 LSpanclspn 19737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lmhm 19788 df-lbs 19841 df-sra 19938 df-rgmod 19939 df-nzr 20025 df-dsmm 20870 df-frlm 20885 df-uvc 20921 |
This theorem is referenced by: lbslsp 30933 lbsdiflsp0 31017 fedgmul 31022 |
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