Theorem wemaplem1 8451

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 6190	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑧) = (𝑃‘𝑧))
2		fveq1 6190	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑧) = (𝑄‘𝑧))
3	1, 2	breqan12d 4669	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ↔ (𝑃‘𝑧)𝑆(𝑄‘𝑧)))
4		fveq1 6190	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥‘𝑤) = (𝑃‘𝑤))
5		fveq1 6190	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑤) = (𝑄‘𝑤))
6	4, 5	eqeqan12d 2638	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑃‘𝑤) = (𝑄‘𝑤)))
7	6	imbi2d 330	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → ((𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
8	7	ralbidv 2986	. . . . 5 ⊢ ((𝑥 = 𝑃 ∧ 𝑦 = 𝑄) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤))))
9	3, 8	anbi12d 747	. . . 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 3052	. . 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 6191	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑃‘𝑧) = (𝑃‘𝑎))
12		fveq2 6191	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑄‘𝑧) = (𝑄‘𝑎))
13	11, 12	breq12d 4666	. . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑃‘𝑧)𝑆(𝑄‘𝑧) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎)))
14		breq2 4657	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (𝑤𝑅𝑧 ↔ 𝑤𝑅𝑎))
15	14	imbi1d 331	. . . . . . 7 ⊢ (𝑧 = 𝑎 → ((𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
16	15	ralbidv 2986	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤))))
17		breq1 4656	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑤𝑅𝑎 ↔ 𝑏𝑅𝑎))
18		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑃‘𝑤) = (𝑃‘𝑏))
19		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑄‘𝑤) = (𝑄‘𝑏))
20	18, 19	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → ((𝑃‘𝑤) = (𝑄‘𝑤) ↔ (𝑃‘𝑏) = (𝑄‘𝑏)))
21	17, 20	imbi12d 334	. . . . . . 7 ⊢ (𝑤 = 𝑏 → ((𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
22	21	cbvralv 3171	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑎 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))
23	16, 22	syl6bb 276	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑃‘𝑤) = (𝑄‘𝑤)) ↔ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))
24	13, 23	anbi12d 747	. . . 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	cbvrexv 3172	. . 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	syl6bb 276	. 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 4989	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  {copab 4712  ‘cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896

This theorem is referenced by:  wemaplem2  8452  wemaplem3  8453  wemappo  8454  wemapsolem  8455