imasle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > imasle		Structured version Visualization version GIF version

Theorem imasle 17475

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 2730	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
4		eqid 2730	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2730	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		eqid 2730	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7		eqid 2730	. . 3 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
8		eqid 2730	. . 3 ⊢ (·_𝑖‘𝑅) = (·_𝑖‘𝑅)
9		eqid 2730	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10		eqid 2730	. . 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 2730	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
15	1, 2, 12, 13, 3, 14	imasplusg 17469	. . 3 ⊢ (𝜑 → (+_g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩})
16		eqid 2730	. . . 4 ⊢ (._r‘𝑈) = (._r‘𝑈)
17	1, 2, 12, 13, 4, 16	imasmulr 17470	. . 3 ⊢ (𝜑 → (._r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(._r‘𝑅)𝑞))⟩})
18		eqid 2730	. . . 4 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
19	1, 2, 12, 13, 5, 6, 7, 18	imasvsca 17472	. . 3 ⊢ (𝜑 → ( ·_𝑠 ‘𝑈) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·_𝑠 ‘𝑅)𝑞))))
20		eqidd 2731	. . 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 2730	. . . 4 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈)
22	1, 2, 12, 13, 9, 21	imastset 17474	. . 3 ⊢ (𝜑 → (TopSet‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
23		eqid 2730	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
24	1, 2, 12, 13, 10, 23	imasds 17465	. . 3 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑢)) ∣ ((𝐹‘(1^st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(𝑤‘𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2^nd ‘(𝑤‘𝑣))) = (𝐹‘(1^st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ^_𝑠 Σ_g ((dist‘𝑅) ∘ 𝑧))), ℝ^, < )))
25		eqidd 2731	. . 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 17463	. 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 2730	. . 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 17403	. 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 17318	. 2 ⊢ le = Slot (le‘ndx)
30		snsstp2 4821	. . 3 ⊢ {⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ^◡𝐹)⟩} ⊆ {⟨(TopSet‘ndx), (TopSet‘𝑈)⟩, ⟨(le‘ndx), ((𝐹 ∘ 𝑁) ∘ ^◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}
31		ssun2 4174	. . 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 3992	. 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 6806	. . . . . 6 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
34	12, 33	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
35		fvex 6905	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
36	2, 35	eqeltrdi 2839	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
37	34, 36	fexd 7232	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
38	11	fvexi 6906	. . . 4 ⊢ 𝑁 ∈ V
39		coexg 7924	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑁 ∈ V) → (𝐹 ∘ 𝑁) ∈ V)
40	37, 38, 39	sylancl 584	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝑁) ∈ V)
41		cnvexg 7919	. . . 4 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
42	37, 41	syl 17	. . 3 ⊢ (𝜑 → ^◡𝐹 ∈ V)
43		coexg 7924	. . 3 ⊢ (((𝐹 ∘ 𝑁) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ 𝑁) ∘ ^◡𝐹) ∈ V)
44	40, 42, 43	syl2anc 582	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝑁) ∘ ^◡𝐹) ∈ V)
45		imasle.l	. 2 ⊢ ≤ = (le‘𝑈)
46	26, 28, 29, 32, 44, 45	strfv3 17144	1 ⊢ (𝜑 → ≤ = ((𝐹 ∘ 𝑁) ∘ ^◡𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3947 {csn 4629 {ctp 4633 ⟨cop 4635 ∪ ciun 4998 ^◡ccnv 5676 ∘ ccom 5681 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7413 1c1 11115 2c2 12273 cdc 12683 ndxcnx 17132 Basecbs 17150 +_gcplusg 17203 ._rcmulr 17204 Scalarcsca 17206 ·_𝑠 cvsca 17207 ·_𝑖cip 17208 TopSetcts 17209 lecple 17210 distcds 17212 TopOpenctopn 17373 “_s cimas 17456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 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-tp 4634 df-op 4636 df-uni 4910 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-fz 13491 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-imas 17460

This theorem is referenced by: imasless 17492 imasleval 17493

W3C validator