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Theorem spansncol 31548

Description: The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

**Assertion**
Ref	Expression
spansncol	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·_ℎ 𝐴)}) = (span‘{𝐴}))

Proof of Theorem spansncol Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 11090	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
2	1	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
3	2	adantll 714	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
4		ax-hvmulass 30987	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
5	4	3com13 1124	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
6	5	3expa 1118	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
7	6	eqeq2d 2742	. . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴) ↔ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
8	7	biimprd 248	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)))
9		oveq1 7353	. . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·_ℎ 𝐴) = ((𝑦 · 𝐵) ·_ℎ 𝐴))
10	9	rspceeqv 3595	. . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴))
11	3, 8, 10	syl6an 684	. . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
12	11	rexlimdva 3133	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
13	12	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
14		divcl 11782	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ)
15	14	3expb 1120	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
16	15	adantlr 715	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
17		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ)
18		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ)
19		ax-hvmulass 30987	. . . . . . . . . . . . 13 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
20	16, 17, 18, 19	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
21		divcan1 11785	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
22	21	3expb 1120	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
23	22	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
24	23	oveq1d 7361	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = (𝑧 ·_ℎ 𝐴))
25	20, 24	eqtr3d 2768	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)) = (𝑧 ·_ℎ 𝐴))
26	25	eqeq2d 2742	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ 𝑥 = (𝑧 ·_ℎ 𝐴)))
27	26	biimprd 248	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))))
28		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
29	28	rspceeqv 3595	. . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
30	16, 27, 29	syl6an 684	. . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
31	30	exp43 436	. . . . . . 7 ⊢ (𝑧 ∈ ℂ → (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))))
32	31	com4l 92	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))))
33	32	3imp 1110	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))
34	33	rexlimdv 3131	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
35	13, 34	impbid 212	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
36		hvmulcl 30993	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
37	36	ancoms 458	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
38		elspansn 31546	. . . . 5 ⊢ ((𝐵 ·_ℎ 𝐴) ∈ ℋ → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
39	37, 38	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
40	39	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
41		elspansn 31546	. . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
42	41	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
43	35, 40, 42	3bitr4d 311	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ 𝑥 ∈ (span‘{𝐴})))
44	43	eqrdv 2729	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·_ℎ 𝐴)}) = (span‘{𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 {csn 4573 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 0cc0 11006 · cmul 11011 / cdiv 11774 ℋchba 30899 ·_ℎ csm 30901 spancspn 30912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086 ax-hilex 30979 ax-hfvadd 30980 ax-hvcom 30981 ax-hvass 30982 ax-hv0cl 30983 ax-hvaddid 30984 ax-hfvmul 30985 ax-hvmulid 30986 ax-hvmulass 30987 ax-hvdistr1 30988 ax-hvdistr2 30989 ax-hvmul0 30990 ax-hfi 31059 ax-his1 31062 ax-his2 31063 ax-his3 31064 ax-his4 31065 ax-hcompl 31182

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-cn 23142 df-cnp 23143 df-lm 23144 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cfil 25182 df-cau 25183 df-cmet 25184 df-grpo 30473 df-gid 30474 df-ginv 30475 df-gdiv 30476 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-vs 30579 df-nmcv 30580 df-ims 30581 df-dip 30681 df-ssp 30702 df-ph 30793 df-cbn 30843 df-hnorm 30948 df-hba 30949 df-hvsub 30951 df-hlim 30952 df-hcau 30953 df-sh 31187 df-ch 31201 df-oc 31232 df-ch0 31233 df-span 31289

This theorem is referenced by: spansneleq 31550 superpos 32334

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