Theorem wemaplem1 9235

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 6755	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑧) = (𝑃‘𝑧))
2		fveq1 6755	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑧) = (𝑄‘𝑧))
3	1, 2	breqan12d 5086	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ↔ (𝑃‘𝑧)𝑆(𝑄‘𝑧)))
4		fveq1 6755	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑤) = (𝑃‘𝑤))
5		fveq1 6755	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑤) = (𝑄‘𝑤))
6	4, 5	eqeqan12d 2752	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑃‘𝑤) = (𝑄‘𝑤)))
7	6	imbi2d 340	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
8	7	ralbidv 3120	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
9	3, 8	anbi12d 630	. . . 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 3225	. . 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 6756	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑃‘𝑧) = (𝑃‘𝑎))
12		fveq2 6756	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑄‘𝑧) = (𝑄‘𝑎))
13	11, 12	breq12d 5083	. . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎)))
14		breq2 5074	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (𝑤𝑅𝑧 ↔ 𝑤𝑅𝑎))
15	14	imbi1d 341	. . . . . . 7 ⊢ (𝑧 = 𝑎 → ((𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
16	15	ralbidv 3120	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
17		breq1 5073	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑤𝑅𝑎 ↔ 𝑏𝑅𝑎))
18		fveq2 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑃‘𝑤) = (𝑃‘𝑏))
19		fveq2 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑄‘𝑤) = (𝑄‘𝑏))
20	18, 19	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → ((𝑃‘𝑤) = (𝑄‘𝑤) ↔ (𝑃‘𝑏) = (𝑄‘𝑏)))
21	17, 20	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = 𝑏 → ((𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
22	21	cbvralvw 3372	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))
23	16, 22	bitrdi 286	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
24	13, 23	anbi12d 630	. . . 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 3373	. . 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 286	. 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 5440	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   class class class wbr 5070  {copab 5132  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426

This theorem is referenced by:  wemaplem2  9236  wemaplem3  9237  wemappo  9238  wemapsolem  9239