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Theorem spanuni 31064
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 30856 . . . . . . 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 30856 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ (spanβ€˜π΅) ∈ Sβ„‹ )
64, 5ax-mp 5 . . . . . 6 (spanβ€˜π΅) ∈ Sβ„‹
73, 6shscli 30837 . . . . 5 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ∈ Sβ„‹
87shssii 30733 . . . 4 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† β„‹
9 spanss2 30865 . . . . . . 7 (𝐴 βŠ† β„‹ β†’ 𝐴 βŠ† (spanβ€˜π΄))
101, 9ax-mp 5 . . . . . 6 𝐴 βŠ† (spanβ€˜π΄)
11 spanss2 30865 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ 𝐡 βŠ† (spanβ€˜π΅))
124, 11ax-mp 5 . . . . . 6 𝐡 βŠ† (spanβ€˜π΅)
13 unss12 4181 . . . . . 6 ((𝐴 βŠ† (spanβ€˜π΄) ∧ 𝐡 βŠ† (spanβ€˜π΅)) β†’ (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅)))
1410, 12, 13mp2an 688 . . . . 5 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅))
153, 6shunssi 30888 . . . . 5 ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
1614, 15sstri 3990 . . . 4 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
17 spanss 30868 . . . 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 688 . . 3 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)))
19 spanid 30867 . . . 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 4017 . 2 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
223, 6shseli 30836 . . . . 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 274 . . . 4 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ↔ βˆƒπ‘§βˆƒπ‘€((𝑧 ∈ (spanβ€˜π΄) ∧ 𝑀 ∈ (spanβ€˜π΅)) ∧ π‘₯ = (𝑧 +β„Ž 𝑀)))
25 vex 3476 . . . . . . . . . . 11 𝑧 ∈ V
2625elspani 31063 . . . . . . . . . 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 3476 . . . . . . . . . . 11 𝑀 ∈ V
2928elspani 31063 . . . . . . . . . 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 625 . . . . . . . 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 3109 . . . . . . . 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 277 . . . . . . 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 579 . . . . . 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 4183 . . . . . . . . . . . 12 ((𝐴 βŠ† 𝑦 ∧ 𝐡 βŠ† 𝑦) ↔ (𝐴 βˆͺ 𝐡) βŠ† 𝑦)
37 anim12 805 . . . . . . . . . . . 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 30737 . . . . . . . . . . . 12 ((𝑦 ∈ Sβ„‹ ∧ 𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦)
40393expib 1120 . . . . . . . . . . 11 (𝑦 ∈ Sβ„‹ β†’ ((𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
4138, 40sylan9r 507 . . . . . . . . . 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 2819 . . . . . . . . . . 11 (π‘₯ = (𝑧 +β„Ž 𝑀) β†’ (π‘₯ ∈ 𝑦 ↔ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
4342biimprd 247 . . . . . . . . . 10 (π‘₯ = (𝑧 +β„Ž 𝑀) β†’ ((𝑧 +β„Ž 𝑀) ∈ 𝑦 β†’ π‘₯ ∈ 𝑦))
4441, 43sylan9 506 . . . . . . . . 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 456 . . . . . . . 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 3078 . . . . . . 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 4184 . . . . . . . 8 (𝐴 βˆͺ 𝐡) βŠ† β„‹
48 vex 3476 . . . . . . . . 9 π‘₯ ∈ V
4948elspani 31063 . . . . . . . 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 1933 . . . 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 3985 . 2 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† (spanβ€˜(𝐴 βˆͺ 𝐡))
5621, 55eqssi 3997 1 (spanβ€˜(𝐴 βˆͺ 𝐡)) = ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βˆͺ cun 3945   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411   β„‹chba 30439   +β„Ž cva 30440   Sβ„‹ csh 30448   +β„‹ cph 30451  spancspn 30452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192  ax-hilex 30519  ax-hfvadd 30520  ax-hvcom 30521  ax-hvass 30522  ax-hv0cl 30523  ax-hvaddid 30524  ax-hfvmul 30525  ax-hvmulid 30526  ax-hvmulass 30527  ax-hvdistr1 30528  ax-hvdistr2 30529  ax-hvmul0 30530  ax-hfi 30599  ax-his1 30602  ax-his2 30603  ax-his3 30604  ax-his4 30605
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-icc 13335  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-top 22616  df-topon 22633  df-bases 22669  df-lm 22953  df-haus 23039  df-grpo 30013  df-gid 30014  df-ginv 30015  df-gdiv 30016  df-ablo 30065  df-vc 30079  df-nv 30112  df-va 30115  df-ba 30116  df-sm 30117  df-0v 30118  df-vs 30119  df-nmcv 30120  df-ims 30121  df-hnorm 30488  df-hvsub 30491  df-hlim 30492  df-sh 30727  df-ch 30741  df-ch0 30773  df-shs 30828  df-span 30829
This theorem is referenced by:  spanun  31065  spanunsni  31099  spansnji  31166
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