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Theorem spanuni 30528
Description: The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
spanun.1 𝐴 βŠ† β„‹
spanun.2 𝐡 βŠ† β„‹
Assertion
Ref Expression
spanuni (spanβ€˜(𝐴 βˆͺ 𝐡)) = ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))

Proof of Theorem spanuni
Dummy variables π‘₯ 𝑦 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spanun.1 . . . . . . 7 𝐴 βŠ† β„‹
2 spancl 30320 . . . . . . 7 (𝐴 βŠ† β„‹ β†’ (spanβ€˜π΄) ∈ Sβ„‹ )
31, 2ax-mp 5 . . . . . 6 (spanβ€˜π΄) ∈ Sβ„‹
4 spanun.2 . . . . . . 7 𝐡 βŠ† β„‹
5 spancl 30320 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ (spanβ€˜π΅) ∈ Sβ„‹ )
64, 5ax-mp 5 . . . . . 6 (spanβ€˜π΅) ∈ Sβ„‹
73, 6shscli 30301 . . . . 5 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ∈ Sβ„‹
87shssii 30197 . . . 4 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† β„‹
9 spanss2 30329 . . . . . . 7 (𝐴 βŠ† β„‹ β†’ 𝐴 βŠ† (spanβ€˜π΄))
101, 9ax-mp 5 . . . . . 6 𝐴 βŠ† (spanβ€˜π΄)
11 spanss2 30329 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ 𝐡 βŠ† (spanβ€˜π΅))
124, 11ax-mp 5 . . . . . 6 𝐡 βŠ† (spanβ€˜π΅)
13 unss12 4147 . . . . . 6 ((𝐴 βŠ† (spanβ€˜π΄) ∧ 𝐡 βŠ† (spanβ€˜π΅)) β†’ (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅)))
1410, 12, 13mp2an 691 . . . . 5 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅))
153, 6shunssi 30352 . . . . 5 ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
1614, 15sstri 3958 . . . 4 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
17 spanss 30332 . . . 4 ((((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† β„‹ ∧ (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))) β†’ (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))))
188, 16, 17mp2an 691 . . 3 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)))
19 spanid 30331 . . . 4 (((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ∈ Sβ„‹ β†’ (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))) = ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)))
207, 19ax-mp 5 . . 3 (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))) = ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
2118, 20sseqtri 3985 . 2 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
223, 6shseli 30300 . . . . 5 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ↔ βˆƒπ‘§ ∈ (spanβ€˜π΄)βˆƒπ‘€ ∈ (spanβ€˜π΅)π‘₯ = (𝑧 +β„Ž 𝑀))
23 r2ex 3193 . . . . 5 (βˆƒπ‘§ ∈ (spanβ€˜π΄)βˆƒπ‘€ ∈ (spanβ€˜π΅)π‘₯ = (𝑧 +β„Ž 𝑀) ↔ βˆƒπ‘§βˆƒπ‘€((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)))
2422, 23bitri 275 . . . 4 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ↔ βˆƒπ‘§βˆƒπ‘€((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)))
25 vex 3452 . . . . . . . . . . 11 𝑧 ∈ V
2625elspani 30527 . . . . . . . . . 10 (𝐴 βŠ† β„‹ β†’ (𝑧 ∈ (spanβ€˜π΄) ↔ βˆ€π‘¦ ∈ Sβ„‹ (𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦)))
271, 26ax-mp 5 . . . . . . . . 9 (𝑧 ∈ (spanβ€˜π΄) ↔ βˆ€π‘¦ ∈ Sβ„‹ (𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦))
28 vex 3452 . . . . . . . . . . 11 𝑀 ∈ V
2928elspani 30527 . . . . . . . . . 10 (𝐡 βŠ† β„‹ β†’ (𝑀 ∈ (spanβ€˜π΅) ↔ βˆ€π‘¦ ∈ Sβ„‹ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)))
304, 29ax-mp 5 . . . . . . . . 9 (𝑀 ∈ (spanβ€˜π΅) ↔ βˆ€π‘¦ ∈ Sβ„‹ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦))
3127, 30anbi12i 628 . . . . . . . 8 ((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ↔ (βˆ€π‘¦ ∈ Sβ„‹ (𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ βˆ€π‘¦ ∈ Sβ„‹ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)))
32 r19.26 3115 . . . . . . . 8 (βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ↔ (βˆ€π‘¦ ∈ Sβ„‹ (𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ βˆ€π‘¦ ∈ Sβ„‹ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)))
3331, 32bitr4i 278 . . . . . . 7 ((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ↔ βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)))
34 r19.27v 3185 . . . . . . 7 ((βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ βˆ€π‘¦ ∈ Sβ„‹ (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)))
3533, 34sylanb 582 . . . . . 6 (((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ βˆ€π‘¦ ∈ Sβ„‹ (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)))
36 unss 4149 . . . . . . . . . . . 12 ((𝐴 βŠ† 𝑦 ∧ 𝐡 βŠ† 𝑦) ↔ (𝐴 βˆͺ 𝐡) βŠ† 𝑦)
37 anim12 808 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) β†’ ((𝐴 βŠ† 𝑦 ∧ 𝐡 βŠ† 𝑦) β†’ (𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦)))
3836, 37biimtrrid 242 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) β†’ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ (𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦)))
39 shaddcl 30201 . . . . . . . . . . . 12 ((𝑦 ∈ Sβ„‹ ∧ 𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦)
40393expib 1123 . . . . . . . . . . 11 (𝑦 ∈ Sβ„‹ β†’ ((𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
4138, 40sylan9r 510 . . . . . . . . . 10 ((𝑦 ∈ Sβ„‹ ∧ ((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦))) β†’ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
42 eleq1 2826 . . . . . . . . . . 11 (π‘₯ = (𝑧 +β„Ž 𝑀) β†’ (π‘₯ ∈ 𝑦 ↔ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
4342biimprd 248 . . . . . . . . . 10 (π‘₯ = (𝑧 +β„Ž 𝑀) β†’ ((𝑧 +β„Ž 𝑀) ∈ 𝑦 β†’ π‘₯ ∈ 𝑦))
4441, 43sylan9 509 . . . . . . . . 9 (((𝑦 ∈ Sβ„‹ ∧ ((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦))) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ π‘₯ ∈ 𝑦))
4544expl 459 . . . . . . . 8 (𝑦 ∈ Sβ„‹ β†’ ((((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ π‘₯ ∈ 𝑦)))
4645ralimia 3084 . . . . . . 7 (βˆ€π‘¦ ∈ Sβ„‹ (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ π‘₯ ∈ 𝑦))
471, 4unssi 4150 . . . . . . . 8 (𝐴 βˆͺ 𝐡) βŠ† β„‹
48 vex 3452 . . . . . . . . 9 π‘₯ ∈ V
4948elspani 30527 . . . . . . . 8 ((𝐴 βˆͺ 𝐡) βŠ† β„‹ β†’ (π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)) ↔ βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ π‘₯ ∈ 𝑦)))
5047, 49ax-mp 5 . . . . . . 7 (π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)) ↔ βˆ€π‘¦ ∈ Sβ„‹ ((𝐴 βˆͺ 𝐡) βŠ† 𝑦 β†’ π‘₯ ∈ 𝑦))
5146, 50sylibr 233 . . . . . 6 (βˆ€π‘¦ ∈ Sβ„‹ (((𝐴 βŠ† 𝑦 β†’ 𝑧 ∈ 𝑦) ∧ (𝐡 βŠ† 𝑦 β†’ 𝑀 ∈ 𝑦)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)))
5235, 51syl 17 . . . . 5 (((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)))
5352exlimivv 1936 . . . 4 (βˆƒπ‘§βˆƒπ‘€((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)) β†’ π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)))
5424, 53sylbi 216 . . 3 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) β†’ π‘₯ ∈ (spanβ€˜(𝐴 βˆͺ 𝐡)))
5554ssriv 3953 . 2 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† (spanβ€˜(𝐴 βˆͺ 𝐡))
5621, 55eqssi 3965 1 (spanβ€˜(𝐴 βˆͺ 𝐡)) = ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074   βˆͺ cun 3913   βŠ† wss 3915  β€˜cfv 6501  (class class class)co 7362   β„‹chba 29903   +β„Ž cva 29904   Sβ„‹ csh 29912   +β„‹ cph 29915  spancspn 29916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138  ax-hilex 29983  ax-hfvadd 29984  ax-hvcom 29985  ax-hvass 29986  ax-hv0cl 29987  ax-hvaddid 29988  ax-hfvmul 29989  ax-hvmulid 29990  ax-hvmulass 29991  ax-hvdistr1 29992  ax-hvdistr2 29993  ax-hvmul0 29994  ax-hfi 30063  ax-his1 30066  ax-his2 30067  ax-his3 30068  ax-his4 30069
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-icc 13278  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-topgen 17332  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-lm 22596  df-haus 22682  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480  df-ablo 29529  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-vs 29583  df-nmcv 29584  df-ims 29585  df-hnorm 29952  df-hvsub 29955  df-hlim 29956  df-sh 30191  df-ch 30205  df-ch0 30237  df-shs 30292  df-span 30293
This theorem is referenced by:  spanun  30529  spanunsni  30563  spansnji  30630
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