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Theorem spanuni 30784
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 30576 . . . . . . 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 30576 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ (spanβ€˜π΅) ∈ Sβ„‹ )
64, 5ax-mp 5 . . . . . 6 (spanβ€˜π΅) ∈ Sβ„‹
73, 6shscli 30557 . . . . 5 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ∈ Sβ„‹
87shssii 30453 . . . 4 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† β„‹
9 spanss2 30585 . . . . . . 7 (𝐴 βŠ† β„‹ β†’ 𝐴 βŠ† (spanβ€˜π΄))
101, 9ax-mp 5 . . . . . 6 𝐴 βŠ† (spanβ€˜π΄)
11 spanss2 30585 . . . . . . 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 690 . . . . 5 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅))
153, 6shunssi 30608 . . . . 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 30588 . . . 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 690 . . 3 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (spanβ€˜((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)))
19 spanid 30587 . . . 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 30556 . . . . 5 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ↔ βˆƒπ‘§ ∈ (spanβ€˜π΄)βˆƒπ‘€ ∈ (spanβ€˜π΅)π‘₯ = (𝑧 +β„Ž 𝑀))
23 r2ex 3195 . . . . 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 3478 . . . . . . . . . . 11 𝑧 ∈ V
2625elspani 30783 . . . . . . . . . 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 3478 . . . . . . . . . . 11 𝑀 ∈ V
2928elspani 30783 . . . . . . . . . 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 627 . . . . . . . 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 3111 . . . . . . . 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 3187 . . . . . . 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 581 . . . . . 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 807 . . . . . . . . . . . 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 30457 . . . . . . . . . . . 12 ((𝑦 ∈ Sβ„‹ ∧ 𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦)
40393expib 1122 . . . . . . . . . . 11 (𝑦 ∈ Sβ„‹ β†’ ((𝑧 ∈ 𝑦 ∧ 𝑀 ∈ 𝑦) β†’ (𝑧 +β„Ž 𝑀) ∈ 𝑦))
4138, 40sylan9r 509 . . . . . . . . . 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 2821 . . . . . . . . . . 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 508 . . . . . . . . 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 458 . . . . . . . 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 3080 . . . . . . 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 3478 . . . . . . . . 9 π‘₯ ∈ V
4948elspani 30783 . . . . . . . 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 1935 . . . 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βˆͺ cun 3945   βŠ† wss 3947  β€˜cfv 6540  (class class class)co 7405   β„‹chba 30159   +β„Ž cva 30160   Sβ„‹ csh 30168   +β„‹ cph 30171  spancspn 30172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186  ax-hilex 30239  ax-hfvadd 30240  ax-hvcom 30241  ax-hvass 30242  ax-hv0cl 30243  ax-hvaddid 30244  ax-hfvmul 30245  ax-hvmulid 30246  ax-hvmulass 30247  ax-hvdistr1 30248  ax-hvdistr2 30249  ax-hvmul0 30250  ax-hfi 30319  ax-his1 30322  ax-his2 30323  ax-his3 30324  ax-his4 30325
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-icc 13327  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-lm 22724  df-haus 22810  df-grpo 29733  df-gid 29734  df-ginv 29735  df-gdiv 29736  df-ablo 29785  df-vc 29799  df-nv 29832  df-va 29835  df-ba 29836  df-sm 29837  df-0v 29838  df-vs 29839  df-nmcv 29840  df-ims 29841  df-hnorm 30208  df-hvsub 30211  df-hlim 30212  df-sh 30447  df-ch 30461  df-ch0 30493  df-shs 30548  df-span 30549
This theorem is referenced by:  spanun  30785  spanunsni  30819  spansnji  30886
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