ellspds - Mathbox for Thierry Arnoux

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Theorem ellspds 31609

Description: Variation on ellspd 21054. (Contributed by Thierry Arnoux, 18-May-2023.)

**Hypotheses**
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 = (0_g‘𝑆)
ellspds.t	⊢ · = ( ·_𝑠 ‘𝑀)
ellspds.m	⊢ (𝜑 → 𝑀 ∈ LMod)
ellspds.1	⊢ (𝜑 → 𝑉 ⊆ 𝐵)

**Assertion**
Ref	Expression
ellspds	⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))))

Distinct variable groups: 𝐵,𝑎 𝐾,𝑎,𝑣 𝑁,𝑎 𝑀,𝑎 𝑆,𝑎 𝑋,𝑎 𝑉,𝑎,𝑣 𝜑,𝑎,𝑣 0 ,𝑎 · ,𝑎,𝑣 Allowed substitution hints: 𝐵(𝑣) 𝑆(𝑣) 𝑀(𝑣) 𝑁(𝑣) 𝑋(𝑣) 0 (𝑣)

**Proof of Theorem ellspds**
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 = (0_g‘𝑆)
6		ellspds.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑀)
7		f1oi 6784	. . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
8		f1of 6746	. . . . 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 6648	. . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵)
12		ellspds.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
13	2	fvexi 6818	. . . . 5 ⊢ 𝐵 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
15	14, 10	ssexd 5257	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
16	1, 2, 3, 4, 5, 6, 11, 12, 15	ellspd 21054	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))))))
17		ssid 3948	. . . . 5 ⊢ 𝑉 ⊆ 𝑉
18		resiima 5994	. . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
20	19	fveq2d 6808	. . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉))
21	20	eleq2d 2822	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉)))
22		elmapfn 8684	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝑉) → 𝑎 Fn 𝑉)
23	22	adantl 483	. . . . . . 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 6631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉)
26	15	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → 𝑉 ∈ V)
27		inidm 4158	. . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉
28		eqidd 2737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣))
29		fvresi 7077	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣)
30	29	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣)
31	23, 25, 26, 26, 27, 28, 30	offval 7574	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑎 ∘_f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))
32	31	oveq2d 7323	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))
33	32	eqeq2d 2747	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) ↔ 𝑋 = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))
34	33	anbi2d 630	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ((𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉)))) ↔ (𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))))
35	34	rexbidva 3170	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉)))) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (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 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ∃wrex 3071 Vcvv 3437 ⊆ wss 3892 class class class wbr 5081 ↦ cmpt 5164 I cid 5499 ↾ cres 5602 “ cima 5603 Fn wfn 6453 ⟶wf 6454 –1-1-onto→wf1o 6457 ‘cfv 6458 (class class class)co 7307 ∘_f cof 7563 ↑_m cmap 8646 finSupp cfsupp 9172 Basecbs 16957 Scalarcsca 17010 ·_𝑠 cvsca 17011 0_gc0g 17195 Σ_g cgsu 17196 LModclmod 20168 LSpanclspn 20278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-sup 9245 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-fzo 13429 df-seq 13768 df-hash 14091 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-hom 17031 df-cco 17032 df-0g 17197 df-gsum 17198 df-prds 17203 df-pws 17205 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-mhm 18475 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-mulg 18746 df-subg 18797 df-ghm 18877 df-cntz 18968 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-subrg 20067 df-lmod 20170 df-lss 20239 df-lsp 20279 df-lmhm 20329 df-lbs 20382 df-sra 20479 df-rgmod 20480 df-nzr 20574 df-dsmm 20984 df-frlm 20999 df-uvc 21035

This theorem is referenced by: elrsp 31614 lbslsp 31617 lbsdiflsp0 31752 fedgmul 31757

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