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Theorem spansncol 31498

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 11233	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
2	1	ancoms 457	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
3	2	adantll 712	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ)
4		ax-hvmulass 30937	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
5	4	3com13 1121	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
6	5	3expa 1115	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·_ℎ 𝐴) = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
7	6	eqeq2d 2737	. . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴) ↔ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
8	7	biimprd 247	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)))
9		oveq1 7423	. . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·_ℎ 𝐴) = ((𝑦 · 𝐵) ·_ℎ 𝐴))
10	9	rspceeqv 3629	. . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴))
11	3, 8, 10	syl6an 682	. . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
12	11	rexlimdva 3145	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
13	12	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
14		divcl 11920	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ)
15	14	3expb 1117	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
16	15	adantlr 713	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ)
17		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ)
18		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ)
19		ax-hvmulass 30937	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·_ℎ 𝐴) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
21		divcan1 11923	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
22	21	3expb 1117	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
23	22	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧)
24	23	oveq1d 7431	. . . . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ 𝑥 = (𝑧 ·_ℎ 𝐴)))
27	26	biimprd 247	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))))
28		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴)))
29	28	rspceeqv 3629	. . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·_ℎ (𝐵 ·_ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))
30	16, 27, 29	syl6an 682	. . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
31	30	exp43 435	. . . . . . 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 1108	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·_ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)))))
34	33	rexlimdv 3143	. . . 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 211	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴)) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
36		hvmulcl 30943	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
37	36	ancoms 457	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·_ℎ 𝐴) ∈ ℋ)
38		elspansn 31496	. . . . 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 1129	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·_ℎ (𝐵 ·_ℎ 𝐴))))
41		elspansn 31496	. . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
42	41	3ad2ant1 1130	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·_ℎ 𝐴)))
43	35, 40, 42	3bitr4d 310	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·_ℎ 𝐴)}) ↔ 𝑥 ∈ (span‘{𝐴})))
44	43	eqrdv 2724	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·_ℎ 𝐴)}) = (span‘{𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 {csn 4623 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 0cc0 11149 · cmul 11154 / cdiv 11912 ℋchba 30849 ·_ℎ csm 30851 spancspn 30862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cc 10469 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229 ax-hilex 30929 ax-hfvadd 30930 ax-hvcom 30931 ax-hvass 30932 ax-hv0cl 30933 ax-hvaddid 30934 ax-hfvmul 30935 ax-hvmulid 30936 ax-hvmulass 30937 ax-hvdistr1 30938 ax-hvdistr2 30939 ax-hvmul0 30940 ax-hfi 31009 ax-his1 31012 ax-his2 31013 ax-his3 31014 ax-his4 31015 ax-hcompl 31132

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-omul 8493 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-acn 9978 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-rlim 15486 df-sum 15686 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-cn 23219 df-cnp 23220 df-lm 23221 df-haus 23307 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cfil 25271 df-cau 25272 df-cmet 25273 df-grpo 30423 df-gid 30424 df-ginv 30425 df-gdiv 30426 df-ablo 30475 df-vc 30489 df-nv 30522 df-va 30525 df-ba 30526 df-sm 30527 df-0v 30528 df-vs 30529 df-nmcv 30530 df-ims 30531 df-dip 30631 df-ssp 30652 df-ph 30743 df-cbn 30793 df-hnorm 30898 df-hba 30899 df-hvsub 30901 df-hlim 30902 df-hcau 30903 df-sh 31137 df-ch 31151 df-oc 31182 df-ch0 31183 df-span 31239

This theorem is referenced by: spansneleq 31500 superpos 32284

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