ellspds - Mathbox for Thierry Arnoux

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Theorem ellspds 33396

Description: Variation on ellspd 21822. (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 6886	. . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
8		f1of 6848	. . . . 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 6753	. . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵)
12		ellspds.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
13	2	fvexi 6920	. . . . 5 ⊢ 𝐵 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
15	14, 10	ssexd 5324	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
16	1, 2, 3, 4, 5, 6, 11, 12, 15	ellspd 21822	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))))))
17		ssid 4006	. . . . 5 ⊢ 𝑉 ⊆ 𝑉
18		resiima 6094	. . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
20	19	fveq2d 6910	. . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉))
21	20	eleq2d 2827	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉)))
22		elmapfn 8905	. . . . . . . 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 6737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉)
26	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → 𝑉 ∈ V)
27		inidm 4227	. . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉
28		eqidd 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣))
29		fvresi 7193	. . . . . . . 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 7706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑎 ∘_f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))
32	31	oveq2d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))
33	32	eqeq2d 2748	. . . 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 3177	. 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 I cid 5577 ↾ cres 5687 “ cima 5688 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∘_f cof 7695 ↑_m cmap 8866 finSupp cfsupp 9401 Basecbs 17247 Scalarcsca 17300 ·_𝑠 cvsca 17301 0_gc0g 17484 Σ_g cgsu 17485 LModclmod 20858 LSpanclspn 20969

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-nzr 20513 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lbs 21074 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-uvc 21803

This theorem is referenced by: elrsp 33400 lbslsp 33405 lbsdiflsp0 33677 fedgmul 33682 fldextrspunlsplem 33723 fldextrspunlsp 33724

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