ellspds - Mathbox for Thierry Arnoux

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Theorem ellspds 33514

Description: Variation on ellspd 21841. (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 6839	. . . . 5 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
8		f1of 6800	. . . . 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 6703	. . 3 ⊢ (𝜑 → ( I ↾ 𝑉):𝑉⟶𝐵)
12		ellspds.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
13	2	fvexi 6875	. . . . 5 ⊢ 𝐵 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
15	14, 10	ssexd 5277	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
16	1, 2, 3, 4, 5, 6, 11, 12, 15	ellspd 21841	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ ∃𝑎 ∈ (𝐾 ↑_m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))))))
17		ssid 3956	. . . . 5 ⊢ 𝑉 ⊆ 𝑉
18		resiima 6060	. . . . 5 ⊢ (𝑉 ⊆ 𝑉 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑉) “ 𝑉) = 𝑉)
20	19	fveq2d 6865	. . 3 ⊢ (𝜑 → (𝑁‘(( I ↾ 𝑉) “ 𝑉)) = (𝑁‘𝑉))
21	20	eleq2d 2847	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(( I ↾ 𝑉) “ 𝑉)) ↔ 𝑋 ∈ (𝑁‘𝑉)))
22		elmapfn 8839	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝑉) → 𝑎 Fn 𝑉)
23	22	adantl 485	. . . . . . 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 6686	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → ( I ↾ 𝑉) Fn 𝑉)
26	15	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → 𝑉 ∈ V)
27		inidm 4176	. . . . . . 7 ⊢ (𝑉 ∩ 𝑉) = 𝑉
28		eqidd 2762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (𝑎‘𝑣) = (𝑎‘𝑣))
29		fvresi 7151	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → (( I ↾ 𝑉)‘𝑣) = 𝑣)
30	29	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) ∧ 𝑣 ∈ 𝑉) → (( I ↾ 𝑉)‘𝑣) = 𝑣)
31	23, 25, 26, 26, 27, 28, 30	offval 7663	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑎 ∘_f · ( I ↾ 𝑉)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))
32	31	oveq2d 7406	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐾 ↑_m 𝑉)) → (𝑀 Σ_g (𝑎 ∘_f · ( I ↾ 𝑉))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))))
33	32	eqeq2d 2772	. . . 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 639	. . 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 3183	. 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 311	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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∃wrex 3085 Vcvv 3453 ⊆ wss 3902 class class class wbr 5097 ↦ cmpt 5178 I cid 5537 ↾ cres 5645 “ cima 5646 Fn wfn 6510 ⟶wf 6511 –1-1-onto→wf1o 6514 ‘cfv 6515 (class class class)co 7390 ∘_f cof 7652 ↑_m cmap 8801 finSupp cfsupp 9300 Basecbs 17235 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 Σ_g cgsu 17459 LModclmod 20914 LSpanclspn 21025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-fz 13506 df-fzo 13653 df-seq 14008 df-hash 14337 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-nzr 20549 df-subrg 20606 df-lmod 20916 df-lss 20986 df-lsp 21026 df-lmhm 21076 df-lbs 21129 df-sra 21227 df-rgmod 21228 df-dsmm 21771 df-frlm 21786 df-uvc 21822

This theorem is referenced by: elrsp 33518 lbslsp 33523 lbsdiflsp0 33883 fedgmul 33888 fldextrspunlsplem 33930 fldextrspunlsp 33931

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