Theorem spansneleq 5475

Description: Membership relation that implies equality of spans.

Assertion

Ref	Expression
spansneleq	⊢ ((B ∈ ℋ ∧ ¬ A = 0v) → (A ∈ (span ‘{B}) → (span ‘{A}) = (span ‘{B})))

Proof of Theorem spansneleq

Step	Hyp	Ref	Expression
1		elspansnt 5471	. . 3 ⊢ (B ∈ ℋ → (A ∈ (span ‘{B}) ↔ ∃x ∈ ℂ A = (x ·s B)))
2	1	adantr 306	. 2 ⊢ ((B ∈ ℋ ∧ ¬ A = 0v) → (A ∈ (span ‘{B}) ↔ ∃x ∈ ℂ A = (x ·s B)))
3		sneq 1816	. . . . . . 7 ⊢ (A = (x ·s B) → {A} = {(x ·s B)})
4	3	fveq2d 2836	. . . . . 6 ⊢ (A = (x ·s B) → (span ‘{A}) = (span ‘{(x ·s B)}))
5	4	ad2antrr 323	. . . . 5 ⊢ (((B ∈ ℋ ∧ ¬ A = 0v) ∧ (x ∈ ℂ ∧ A = (x ·s B))) → (span ‘{A}) = (span ‘{(x ·s B)}))
6		opreq1 3006	. . . . . . . . . . . . . . . . . 18 ⊢ (x = 0 → (x ·s B) = (0 ·s B))
7		ax-hvmulzer 4995	. . . . . . . . . . . . . . . . . 18 ⊢ (B ∈ ℋ → (0 ·s B) = 0v)
8	6, 7	sylan9eqr 1145	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∈ ℋ ∧ x = 0) → (x ·s B) = 0v)
9	8	exp 291	. . . . . . . . . . . . . . . 16 ⊢ (B ∈ ℋ → (x = 0 → (x ·s B) = 0v))
10		cleq1 1107	. . . . . . . . . . . . . . . . 17 ⊢ (A = (x ·s B) → (A = 0v ↔ (x ·s B) = 0v))
11	10	biimprd 136	. . . . . . . . . . . . . . . 16 ⊢ (A = (x ·s B) → ((x ·s B) = 0v → A = 0v))
12	9, 11	sylan9 359	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ ℋ ∧ A = (x ·s B)) → (x = 0 → A = 0v))
13	12	con3d 87	. . . . . . . . . . . . . 14 ⊢ ((B ∈ ℋ ∧ A = (x ·s B)) → (¬ A = 0v → ¬ x = 0))
14	13	exp 291	. . . . . . . . . . . . 13 ⊢ (B ∈ ℋ → (A = (x ·s B) → (¬ A = 0v → ¬ x = 0)))
15	14	com23 32	. . . . . . . . . . . 12 ⊢ (B ∈ ℋ → (¬ A = 0v → (A = (x ·s B) → ¬ x = 0)))
16	15	imp3a 279	. . . . . . . . . . 11 ⊢ (B ∈ ℋ → ((¬ A = 0v ∧ A = (x ·s B)) → ¬ x = 0))
17		df-ne 1192	. . . . . . . . . . 11 ⊢ (x ≠ 0 ↔ ¬ x = 0)
18	16, 17	syl6ibr 186	. . . . . . . . . 10 ⊢ (B ∈ ℋ → ((¬ A = 0v ∧ A = (x ·s B)) → x ≠ 0))
19	18	adantr 306	. . . . . . . . 9 ⊢ ((B ∈ ℋ ∧ x ∈ ℂ) → ((¬ A = 0v ∧ A = (x ·s B)) → x ≠ 0))
20		spansncol 5473	. . . . . . . . . . 11 ⊢ ((B ∈ ℋ ∧ x ∈ ℂ ∧ x ≠ 0) → (span ‘{(x ·s B)}) = (span ‘{B}))
21	20	3exp 611	. . . . . . . . . 10 ⊢ (B ∈ ℋ → (x ∈ ℂ → (x ≠ 0 → (span ‘{(x ·s B)}) = (span ‘{B}))))
22	21	imp 277	. . . . . . . . 9 ⊢ ((B ∈ ℋ ∧ x ∈ ℂ) → (x ≠ 0 → (span ‘{(x ·s B)}) = (span ‘{B})))
23	19, 22	syld 27	. . . . . . . 8 ⊢ ((B ∈ ℋ ∧ x ∈ ℂ) → ((¬ A = 0v ∧ A = (x ·s B)) → (span ‘{(x ·s B)}) = (span ‘{B})))
24	23	exp4b 296	. . . . . . 7 ⊢ (B ∈ ℋ → (x ∈ ℂ → (¬ A = 0v → (A = (x ·s B) → (span ‘{(x ·s B)}) = (span ‘{B})))))
25	24	com23 32	. . . . . 6 ⊢ (B ∈ ℋ → (¬ A = 0v → (x ∈ ℂ → (A = (x ·s B) → (span ‘{(x ·s B)}) = (span ‘{B})))))
26	25	imp43 288	. . . . 5 ⊢ (((B ∈ ℋ ∧ ¬ A = 0v) ∧ (x ∈ ℂ ∧ A = (x ·s B))) → (span ‘{(x ·s B)}) = (span ‘{B}))
27	5, 26	eqtrd 1128	. . . 4 ⊢ (((B ∈ ℋ ∧ ¬ A = 0v) ∧ (x ∈ ℂ ∧ A = (x ·s B))) → (span ‘{A}) = (span ‘{B}))
28	27	exp32 294	. . 3 ⊢ ((B ∈ ℋ ∧ ¬ A = 0v) → (x ∈ ℂ → (A = (x ·s B) → (span ‘{A}) = (span ‘{B}))))
29	28	r19.23adv 1286	. 2 ⊢ ((B ∈ ℋ ∧ ¬ A = 0v) → (∃x ∈ ℂ A = (x ·s B) → (span ‘{A}) = (span ‘{B})))
30	2, 29	sylbid 178	1 ⊢ ((B ∈ ℋ ∧ ¬ A = 0v) → (A ∈ (span ‘{B}) → (span ‘{A}) = (span ‘{B})))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ≠ wne 1190  ∃wrex 1202  {csn 1808   ‘cfv 2422  (class class class)co 3001  ℂcc 4026  0cc0 4028   ℋ chil 4958   ·_s csm 4960  0_vc0v 4961  spancspn 4971

This theorem is referenced by:  elspansn4t 5478  spansncv 5542

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-clim 4876  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107  df-sh 5114  df-ch 5127  df-oc 5156  df-ch0 5157  df-span 5276

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