Theorem wemaplem1 9518

Description: Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
wemapso.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}

Assertion

Ref	Expression
wemaplem1	⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))

Distinct variable groups:   𝑎,𝑏,𝑥   𝑇,𝑎,𝑏   𝑤,𝑎,𝑦,𝑧,𝑏,𝑥,𝐴   𝑃,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑇(𝑥, 𝑦, 𝑧, 𝑤)   𝑉(𝑥, 𝑦, 𝑧, 𝑤, 𝑎, 𝑏)   𝑊(𝑥, 𝑦, 𝑧, 𝑤, 𝑎, 𝑏)

Proof of Theorem wemaplem1

Step	Hyp	Ref	Expression
1		fveq1 6887	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑧) = (𝑃‘𝑧))
2		fveq1 6887	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑧) = (𝑄‘𝑧))
3	1, 2	breqan12d 5130	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ↔ (𝑃‘𝑧)𝑆(𝑄‘𝑧)))
4		fveq1 6887	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑤) = (𝑃‘𝑤))
5		fveq1 6887	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑤) = (𝑄‘𝑤))
6	4, 5	eqeqan12d 2780	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑃‘𝑤) = (𝑄‘𝑤)))
7	6	imbi2d 343	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
8	7	ralbidv 3191	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
9	3, 8	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)))))
10	9	rexbidv 3192	. . 3 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐴 ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)))))
11		fveq2 6888	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑃‘𝑧) = (𝑃‘𝑎))
12		fveq2 6888	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑄‘𝑧) = (𝑄‘𝑎))
13	11, 12	breq12d 5127	. . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎)))
14		breq2 5118	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (𝑤𝑅𝑧 ↔ 𝑤𝑅𝑎))
15	14	imbi1d 344	. . . . . . 7 ⊢ (𝑧 = 𝑎 → ((𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
16	15	ralbidv 3191	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
17		breq1 5117	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑤𝑅𝑎 ↔ 𝑏𝑅𝑎))
18		fveq2 6888	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑃‘𝑤) = (𝑃‘𝑏))
19		fveq2 6888	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑄‘𝑤) = (𝑄‘𝑏))
20	18, 19	eqeq12d 2782	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → ((𝑃‘𝑤) = (𝑄‘𝑤) ↔ (𝑃‘𝑏) = (𝑄‘𝑏)))
21	17, 20	imbi12d 347	. . . . . . 7 ⊢ (𝑤 = 𝑏 → ((𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
22	21	cbvralvw 3246	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))
23	16, 22	bitrdi 290	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
24	13, 23	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑎 → (((𝑃‘𝑧)𝑆(𝑄‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))) ↔ ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))
25	24	cbvrexvw 3247	. . 3 ⊢ (∃𝑧 ∈ 𝐴 ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))) ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
26	10, 25	bitrdi 290	. 2 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))
27		wemapso.t	. 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
28	26, 27	brabga 5523	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   class class class wbr 5114  {copab 5178  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-iota 6499  df-fv 6551

This theorem is used by:  wemaplem2  9519  wemaplem3  9520  wemappo  9521  wemapsolem  9522