ellspds - Mathbox for Thierry Arnoux

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Theorem ellspds 33458

Description: Variation on ellspd 21784. (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 6812	. . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
8		f1of 6774	. . . . 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 6679	. . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵)
12		ellspds.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
13	2	fvexi 6848	. . . . 5 ⊢ 𝐵 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
15	14, 10	ssexd 5259	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
16	1, 2, 3, 4, 5, 6, 11, 12, 15	ellspd 21784	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))))))
17		ssid 3944	. . . . 5 ⊢ 𝑉 ⊆ 𝑉
18		resiima 6035	. . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
20	19	fveq2d 6838	. . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉))
21	20	eleq2d 2826	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉)))
22		elmapfn 8809	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝑉) → 𝑎 Fn 𝑉)
23	22	adantl 482	. . . . . . 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 6663	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉)
26	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → 𝑉 ∈ V)
27		inidm 4162	. . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉
28		eqidd 2741	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣))
29		fvresi 7124	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣)
30	29	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣)
31	23, 25, 26, 26, 27, 28, 30	offval 7636	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑎 ∘_f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))
32	31	oveq2d 7379	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))
33	32	eqeq2d 2751	. . . 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 636	. . 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 3162	. 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 310	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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∃wrex 3064 Vcvv 3432 ⊆ wss 3890 class class class wbr 5079 ↦ cmpt 5160 I cid 5519 ↾ cres 5627 “ cima 5628 Fn wfn 6487 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 ∘_f cof 7625 ↑_m cmap 8770 finSupp cfsupp 9271 Basecbs 17177 Scalarcsca 17221 ·_𝑠 cvsca 17222 0_gc0g 17400 Σ_g cgsu 17401 LModclmod 20857 LSpanclspn 20968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-fzo 13607 df-seq 13962 df-hash 14291 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17402 df-gsum 17403 df-prds 17408 df-pws 17410 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-ghm 19186 df-cntz 19290 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-nzr 20492 df-subrg 20549 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lmhm 21019 df-lbs 21072 df-sra 21170 df-rgmod 21171 df-dsmm 21714 df-frlm 21729 df-uvc 21765

This theorem is referenced by: elrsp 33462 lbslsp 33467 lbsdiflsp0 33817 fedgmul 33822 fldextrspunlsplem 33864 fldextrspunlsp 33865

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