ellspds - Mathbox for Thierry Arnoux

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Theorem ellspds 33318

Description: Variation on ellspd 21727. (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 6806	. . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
8		f1of 6768	. . . . 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 6673	. . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵)
12		ellspds.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
13	2	fvexi 6840	. . . . 5 ⊢ 𝐵 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
15	14, 10	ssexd 5266	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
16	1, 2, 3, 4, 5, 6, 11, 12, 15	ellspd 21727	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))))))
17		ssid 3960	. . . . 5 ⊢ 𝑉 ⊆ 𝑉
18		resiima 6031	. . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
20	19	fveq2d 6830	. . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉))
21	20	eleq2d 2814	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉)))
22		elmapfn 8799	. . . . . . . 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 6657	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉)
26	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → 𝑉 ∈ V)
27		inidm 4180	. . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉
28		eqidd 2730	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣))
29		fvresi 7113	. . . . . . . 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 7626	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑎 ∘_f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))
32	31	oveq2d 7369	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))
33	32	eqeq2d 2740	. . . 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 3151	. 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 2109 ∃wrex 3053 Vcvv 3438 ⊆ wss 3905 class class class wbr 5095 ↦ cmpt 5176 I cid 5517 ↾ cres 5625 “ cima 5626 Fn wfn 6481 ⟶wf 6482 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 ↑_m cmap 8760 finSupp cfsupp 9270 Basecbs 17138 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 Σ_g cgsu 17362 LModclmod 20781 LSpanclspn 20892

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-nzr 20416 df-subrg 20473 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lmhm 20944 df-lbs 20997 df-sra 21095 df-rgmod 21096 df-dsmm 21657 df-frlm 21672 df-uvc 21708

This theorem is referenced by: elrsp 33322 lbslsp 33327 lbsdiflsp0 33601 fedgmul 33606 fldextrspunlsplem 33647 fldextrspunlsp 33648

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