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Theorem spanuni 30797
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 30589 . . . . . . 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 30589 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ (spanβ€˜π΅) ∈ Sβ„‹ )
64, 5ax-mp 5 . . . . . 6 (spanβ€˜π΅) ∈ Sβ„‹
73, 6shscli 30570 . . . . 5 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ∈ Sβ„‹
87shssii 30466 . . . 4 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† β„‹
9 spanss2 30598 . . . . . . 7 (𝐴 βŠ† β„‹ β†’ 𝐴 βŠ† (spanβ€˜π΄))
101, 9ax-mp 5 . . . . . 6 𝐴 βŠ† (spanβ€˜π΄)
11 spanss2 30598 . . . . . . 7 (𝐡 βŠ† β„‹ β†’ 𝐡 βŠ† (spanβ€˜π΅))
124, 11ax-mp 5 . . . . . 6 𝐡 βŠ† (spanβ€˜π΅)
13 unss12 4183 . . . . . 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 30621 . . . . 5 ((spanβ€˜π΄) βˆͺ (spanβ€˜π΅)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
1614, 15sstri 3992 . . . 4 (𝐴 βˆͺ 𝐡) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
17 spanss 30601 . . . 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 30600 . . . 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 4019 . 2 (spanβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅))
223, 6shseli 30569 . . . . 5 (π‘₯ ∈ ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) ↔ βˆƒπ‘§ ∈ (spanβ€˜π΄)βˆƒπ‘€ ∈ (spanβ€˜π΅)π‘₯ = (𝑧 +β„Ž 𝑀))
23 r2ex 3196 . . . . 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 3479 . . . . . . . . . . 11 𝑧 ∈ V
2625elspani 30796 . . . . . . . . . 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 3479 . . . . . . . . . . 11 𝑀 ∈ V
2928elspani 30796 . . . . . . . . . 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 3112 . . . . . . . 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 3188 . . . . . . 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 4185 . . . . . . . . . . . 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 30470 . . . . . . . . . . . 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 2822 . . . . . . . . . . 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 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 3081 . . . . . . 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 4186 . . . . . . . 8 (𝐴 βˆͺ 𝐡) βŠ† β„‹
48 vex 3479 . . . . . . . . 9 π‘₯ ∈ V
4948elspani 30796 . . . . . . . 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 3987 . 2 ((spanβ€˜π΄) +β„‹ (spanβ€˜π΅)) βŠ† (spanβ€˜(𝐴 βˆͺ 𝐡))
5621, 55eqssi 3999 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 3062  βˆƒwrex 3071   βˆͺ cun 3947   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409   β„‹chba 30172   +β„Ž cva 30173   Sβ„‹ csh 30181   +β„‹ cph 30184  spancspn 30185
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190  ax-hilex 30252  ax-hfvadd 30253  ax-hvcom 30254  ax-hvass 30255  ax-hv0cl 30256  ax-hvaddid 30257  ax-hfvmul 30258  ax-hvmulid 30259  ax-hvmulass 30260  ax-hvdistr1 30261  ax-hvdistr2 30262  ax-hvmul0 30263  ax-hfi 30332  ax-his1 30335  ax-his2 30336  ax-his3 30337  ax-his4 30338
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-icc 13331  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-lm 22733  df-haus 22819  df-grpo 29746  df-gid 29747  df-ginv 29748  df-gdiv 29749  df-ablo 29798  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-vs 29852  df-nmcv 29853  df-ims 29854  df-hnorm 30221  df-hvsub 30224  df-hlim 30225  df-sh 30460  df-ch 30474  df-ch0 30506  df-shs 30561  df-span 30562
This theorem is referenced by:  spanun  30798  spanunsni  30832  spansnji  30899
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