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Theorem imasle 16006
Description: The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasbas.u (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
imasle.n 𝑁 = (le‘𝑅)
imasle.l = (le‘𝑈)
Assertion
Ref Expression
imasle (𝜑 = ((𝐹𝑁) ∘ 𝐹))

Proof of Theorem imasle
Dummy variables 𝑝 𝑞 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2610 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2610 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2610 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
6 eqid 2610 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7 eqid 2610 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
8 eqid 2610 . . 3 (·𝑖𝑅) = (·𝑖𝑅)
9 eqid 2610 . . 3 (TopOpen‘𝑅) = (TopOpen‘𝑅)
10 eqid 2610 . . 3 (dist‘𝑅) = (dist‘𝑅)
11 imasle.n . . 3 𝑁 = (le‘𝑅)
12 imasbas.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
13 imasbas.r . . . 4 (𝜑𝑅𝑍)
14 eqid 2610 . . . 4 (+g𝑈) = (+g𝑈)
151, 2, 12, 13, 3, 14imasplusg 16000 . . 3 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
16 eqid 2610 . . . 4 (.r𝑈) = (.r𝑈)
171, 2, 12, 13, 4, 16imasmulr 16001 . . 3 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
18 eqid 2610 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 16003 . . 3 (𝜑 → ( ·𝑠𝑈) = 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
20 eqidd 2611 . . 3 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
21 eqid 2610 . . . 4 (TopSet‘𝑈) = (TopSet‘𝑈)
221, 2, 12, 13, 9, 21imastset 16005 . . 3 (𝜑 → (TopSet‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
23 eqid 2610 . . . 4 (dist‘𝑈) = (dist‘𝑈)
241, 2, 12, 13, 10, 23imasds 15996 . . 3 (𝜑 → (dist‘𝑈) = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
25 eqidd 2611 . . 3 (𝜑 → ((𝐹𝑁) ∘ 𝐹) = ((𝐹𝑁) ∘ 𝐹))
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 22, 24, 25, 12, 13imasval 15994 . 2 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
27 eqid 2610 . . 3 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
2827imasvalstr 15935 . 2 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, 12⟩
29 pleid 15872 . 2 le = Slot (le‘ndx)
30 snsstp2 4288 . . 3 {⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩} ⊆ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}
31 ssun2 3739 . . 3 {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3230, 31sstri 3577 . 2 {⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
33 fof 6028 . . . . . 6 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
3412, 33syl 17 . . . . 5 (𝜑𝐹:𝑉𝐵)
35 fvex 6113 . . . . . 6 (Base‘𝑅) ∈ V
362, 35syl6eqel 2696 . . . . 5 (𝜑𝑉 ∈ V)
37 fex 6394 . . . . 5 ((𝐹:𝑉𝐵𝑉 ∈ V) → 𝐹 ∈ V)
3834, 36, 37syl2anc 691 . . . 4 (𝜑𝐹 ∈ V)
39 fvex 6113 . . . . 5 (le‘𝑅) ∈ V
4011, 39eqeltri 2684 . . . 4 𝑁 ∈ V
41 coexg 7010 . . . 4 ((𝐹 ∈ V ∧ 𝑁 ∈ V) → (𝐹𝑁) ∈ V)
4238, 40, 41sylancl 693 . . 3 (𝜑 → (𝐹𝑁) ∈ V)
43 cnvexg 7005 . . . 4 (𝐹 ∈ V → 𝐹 ∈ V)
4438, 43syl 17 . . 3 (𝜑𝐹 ∈ V)
45 coexg 7010 . . 3 (((𝐹𝑁) ∈ V ∧ 𝐹 ∈ V) → ((𝐹𝑁) ∘ 𝐹) ∈ V)
4642, 44, 45syl2anc 691 . 2 (𝜑 → ((𝐹𝑁) ∘ 𝐹) ∈ V)
47 imasle.l . 2 = (le‘𝑈)
4826, 28, 29, 32, 46, 47strfv3 15736 1 (𝜑 = ((𝐹𝑁) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  {csn 4125  {ctp 4129  cop 4131   ciun 4455  ccnv 5037  ccom 5042  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  cdc 11369  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  ·𝑖cip 15773  TopSetcts 15774  lecple 15775  distcds 15777  TopOpenctopn 15905  s cimas 15987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-imas 15991
This theorem is referenced by:  imasless  16023  imasleval  16024
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