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Theorem imasle 16115
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 2621 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2621 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2621 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
6 eqid 2621 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7 eqid 2621 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
8 eqid 2621 . . 3 (·𝑖𝑅) = (·𝑖𝑅)
9 eqid 2621 . . 3 (TopOpen‘𝑅) = (TopOpen‘𝑅)
10 eqid 2621 . . 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 2621 . . . 4 (+g𝑈) = (+g𝑈)
151, 2, 12, 13, 3, 14imasplusg 16109 . . 3 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
16 eqid 2621 . . . 4 (.r𝑈) = (.r𝑈)
171, 2, 12, 13, 4, 16imasmulr 16110 . . 3 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
18 eqid 2621 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 16112 . . 3 (𝜑 → ( ·𝑠𝑈) = 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
20 eqidd 2622 . . 3 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
21 eqid 2621 . . . 4 (TopSet‘𝑈) = (TopSet‘𝑈)
221, 2, 12, 13, 9, 21imastset 16114 . . 3 (𝜑 → (TopSet‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
23 eqid 2621 . . . 4 (dist‘𝑈) = (dist‘𝑈)
241, 2, 12, 13, 10, 23imasds 16105 . . 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 2622 . . 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 16103 . 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 2621 . . 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 16044 . 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 15981 . 2 le = Slot (le‘ndx)
30 snsstp2 4321 . . 3 {⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩} ⊆ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹𝑁) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}
31 ssun2 3760 . . 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 3596 . 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 6077 . . . . . 6 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
3412, 33syl 17 . . . . 5 (𝜑𝐹:𝑉𝐵)
35 fvex 6163 . . . . . 6 (Base‘𝑅) ∈ V
362, 35syl6eqel 2706 . . . . 5 (𝜑𝑉 ∈ V)
37 fex 6450 . . . . 5 ((𝐹:𝑉𝐵𝑉 ∈ V) → 𝐹 ∈ V)
3834, 36, 37syl2anc 692 . . . 4 (𝜑𝐹 ∈ V)
39 fvex 6163 . . . . 5 (le‘𝑅) ∈ V
4011, 39eqeltri 2694 . . . 4 𝑁 ∈ V
41 coexg 7071 . . . 4 ((𝐹 ∈ V ∧ 𝑁 ∈ V) → (𝐹𝑁) ∈ V)
4238, 40, 41sylancl 693 . . 3 (𝜑 → (𝐹𝑁) ∈ V)
43 cnvexg 7066 . . . 4 (𝐹 ∈ V → 𝐹 ∈ V)
4438, 43syl 17 . . 3 (𝜑𝐹 ∈ V)
45 coexg 7071 . . 3 (((𝐹𝑁) ∈ V ∧ 𝐹 ∈ V) → ((𝐹𝑁) ∘ 𝐹) ∈ V)
4642, 44, 45syl2anc 692 . 2 (𝜑 → ((𝐹𝑁) ∘ 𝐹) ∈ V)
47 imasle.l . 2 = (le‘𝑈)
4826, 28, 29, 32, 46, 47strfv3 15840 1 (𝜑 = ((𝐹𝑁) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3189  cun 3557  {csn 4153  {ctp 4157  cop 4159   ciun 4490  ccnv 5078  ccom 5083  wf 5848  ontowfo 5850  cfv 5852  (class class class)co 6610  1c1 9889  2c2 11022  cdc 11445  ndxcnx 15789  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Scalarcsca 15876   ·𝑠 cvsca 15877  ·𝑖cip 15878  TopSetcts 15879  lecple 15880  distcds 15882  TopOpenctopn 16014  s cimas 16096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-imas 16100
This theorem is referenced by:  imasless  16132  imasleval  16133
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