Theorem imasle 16183

Description: The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasle.n	⊢ 𝑁 = (le‘𝑅)
imasle.l	⊢ ≤ = (le‘𝑈)

Assertion

Ref	Expression
imasle	⊢ (𝜑 → ≤ = ((𝐹 ∘ 𝑁) ∘ ◡𝐹))

Proof of Theorem imasle

Dummy variables 𝑝 𝑞 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasbas.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2622	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2622	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2622	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		eqid 2622	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7		eqid 2622	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
8		eqid 2622	. . 3 ⊢ (·𝑖‘𝑅) = (·𝑖‘𝑅)
9		eqid 2622	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10		eqid 2622	. . 3 ⊢ (dist‘𝑅) = (dist‘𝑅)
11		imasle.n	. . 3 ⊢ 𝑁 = (le‘𝑅)
12		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
13		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
14		eqid 2622	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15	1, 2, 12, 13, 3, 14	imasplusg 16177	. . 3 ⊢ (𝜑 → (+g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
16		eqid 2622	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
17	1, 2, 12, 13, 4, 16	imasmulr 16178	. . 3 ⊢ (𝜑 → (.r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
18		eqid 2622	. . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
19	1, 2, 12, 13, 5, 6, 7, 18	imasvsca 16180	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑈) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))))
20		eqidd 2623	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩})
21		eqid 2622	. . . 4 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈)
22	1, 2, 12, 13, 9, 21	imastset 16182	. . 3 ⊢ (𝜑 → (TopSet‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
23		eqid 2622	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
24	1, 2, 12, 13, 10, 23	imasds 16173	. . 3 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤‘𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤‘𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
25		eqidd 2623	. . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝑁) ∘ ◡𝐹) = ((𝐹 ∘ 𝑁) ∘ ◡𝐹))
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 22, 24, 25, 12, 13	imasval 16171	. 2 ⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
27		eqid 2622	. . 3 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
28	27	imasvalstr 16112	. 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, ;12⟩
29		pleid 16049	. 2 ⊢ le = Slot (le‘ndx)
30		snsstp2 4348	. . 3 ⊢ {⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩} ⊆ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}
31		ssun2 3777	. . 3 ⊢ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
32	30, 31	sstri 3612	. 2 ⊢ {⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
33		fof 6115	. . . . . 6 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
34	12, 33	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
35		fvex 6201	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
36	2, 35	syl6eqel 2709	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
37		fex 6490	. . . . 5 ⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑉 ∈ V) → 𝐹 ∈ V)
38	34, 36, 37	syl2anc 693	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
39		fvex 6201	. . . . 5 ⊢ (le‘𝑅) ∈ V
40	11, 39	eqeltri 2697	. . . 4 ⊢ 𝑁 ∈ V
41		coexg 7117	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑁 ∈ V) → (𝐹 ∘ 𝑁) ∈ V)
42	38, 40, 41	sylancl 694	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝑁) ∈ V)
43		cnvexg 7112	. . . 4 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
44	38, 43	syl 17	. . 3 ⊢ (𝜑 → ◡𝐹 ∈ V)
45		coexg 7117	. . 3 ⊢ (((𝐹 ∘ 𝑁) ∈ V ∧ ◡𝐹 ∈ V) → ((𝐹 ∘ 𝑁) ∘ ◡𝐹) ∈ V)
46	42, 44, 45	syl2anc 693	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝑁) ∘ ◡𝐹) ∈ V)
47		imasle.l	. 2 ⊢ ≤ = (le‘𝑈)
48	26, 28, 29, 32, 46, 47	strfv3 15908	1 ⊢ (𝜑 → ≤ = ((𝐹 ∘ 𝑁) ∘ ◡𝐹))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  {csn 4177  {ctp 4181  ⟨cop 4183  ∪ ciun 4520  ^◡ccnv 5113   ∘ ccom 5118  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650  1c1 9937  2c2 11070  ;cdc 11493  ndxcnx 15854  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  ·_𝑖cip 15946  TopSetcts 15947  lecple 15948  distcds 15950  TopOpenctopn 16082   “_s cimas 16164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-imas 16168

This theorem is referenced by:  imasless  16200  imasleval  16201