spansncol - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > spansncol		Structured version Visualization version GIF version

Theorem spansncol 31886

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 11183	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
2	1	ancoms 463	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
3	2	adantll 726	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
4		ax-hvmulass 31325	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
5	4	3com13 1140	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
6	5	3expa 1134	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
7	6	eqeq2d 2772	. . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴) ↔ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
8	7	biimprd 251	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)))
9		oveq1 7417	. . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·_ℎ 𝐴) = ((𝑦 · 𝐵) ·_ℎ 𝐴))
10	9	rspceeqv 3603	. . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴))
11	3, 8, 10	syl6an 696	. . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
12	11	rexlimdva 3164	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
13	12	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
14		divcl 11877	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ)
15	14	3expb 1136	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
16	15	adantlr 727	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
17		simprl 782	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ)
18		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ)
19		ax-hvmulass 31325	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
21		divcan1 11880	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
22	21	3expb 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
23	22	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
24	23	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = (𝑧 ·_ℎ 𝐴))
25	20, 24	eqtr3d 2798	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)) = (𝑧 ·_ℎ 𝐴))
26	25	eqeq2d 2772	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ 𝑥 = (𝑧 ·_ℎ 𝐴)))
27	26	biimprd 251	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))))
28		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
29	28	rspceeqv 3603	. . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
30	16, 27, 29	syl6an 696	. . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
31	30	exp43 441	. . . . . . 7 ⊢ (𝑧 ∈ ℂ → (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))))
32	31	com4l 93	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))))
33	32	3imp 1126	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))
34	33	rexlimdv 3162	. . . 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 215	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
36		hvmulcl 31331	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
37	36	ancoms 463	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
38		elspansn 31884	. . . . 5 ⊢ ((𝐵 ·_ℎ 𝐴) ∈ ℋ → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
39	37, 38	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
40	39	3adant3 1148	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
41		elspansn 31884	. . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
42	41	3ad2ant1 1149	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
43	35, 40, 42	3bitr4d 314	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ 𝑥 ∈ (span‘{𝐴})))
44	43	eqrdv 2759	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·_ℎ 𝐴)}) = (span‘{𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3087 {csn 4588 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 0cc0 11099 · cmul 11104 / cdiv 11870 ℋchba 31237 ·_ℎ csm 31239 spancspn 31250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cc 10418 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179 ax-hilex 31317 ax-hfvadd 31318 ax-hvcom 31319 ax-hvass 31320 ax-hv0cl 31321 ax-hvaddid 31322 ax-hfvmul 31323 ax-hvmulid 31324 ax-hvmulass 31325 ax-hvdistr1 31326 ax-hvdistr2 31327 ax-hvmul0 31328 ax-hfi 31397 ax-his1 31400 ax-his2 31401 ax-his3 31402 ax-his4 31403 ax-hcompl 31520

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-omul 8457 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-acn 9927 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-rlim 15539 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-cn 23363 df-cnp 23364 df-lm 23365 df-haus 23451 df-tx 23698 df-hmeo 23891 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-tms 24458 df-cfil 25393 df-cau 25394 df-cmet 25395 df-grpo 30811 df-gid 30812 df-ginv 30813 df-gdiv 30814 df-ablo 30863 df-vc 30877 df-nv 30910 df-va 30913 df-ba 30914 df-sm 30915 df-0v 30916 df-vs 30917 df-nmcv 30918 df-ims 30919 df-dip 31019 df-ssp 31040 df-ph 31131 df-cbn 31181 df-hnorm 31286 df-hba 31287 df-hvsub 31289 df-hlim 31290 df-hcau 31291 df-sh 31525 df-ch 31539 df-oc 31570 df-ch0 31571 df-span 31627

This theorem is referenced by: spansneleq 31888 superpos 32672

W3C validator