Theorem ltpopr 7805

Description: Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7806. (Contributed by Jim Kingdon, 15-Dec-2019.)

Assertion

Ref	Expression
ltpopr	⊢ <P Po P

Proof of Theorem ltpopr

Dummy variables 𝑟 𝑞 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7685	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
2		prdisj 7702	. . . . . . . 8 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
3	1, 2	sylan 283	. . . . . . 7 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
4		ancom 266	. . . . . . 7 ⊢ ((𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)) ↔ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
5	3, 4	sylnib 680	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
6	5	nrexdv 2623	. . . . 5 ⊢ (𝑠 ∈ P → ¬ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
7		ltdfpr 7716	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑠 ∈ P) → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
8	7	anidms 397	. . . . 5 ⊢ (𝑠 ∈ P → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
9	6, 8	mtbird 677	. . . 4 ⊢ (𝑠 ∈ P → ¬ 𝑠<P 𝑠)
10	9	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑠 ∈ P) → ¬ 𝑠<P 𝑠)
11		ltdfpr 7716	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
12	11	3adant3 1041	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
13		ltdfpr 7716	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
14	13	3adant1 1039	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
15	12, 14	anbi12d 473	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) ↔ (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))))
16		reeanv 2701	. . . . . . . . 9 ⊢ (∃𝑞 ∈ Q ∃𝑟 ∈ Q ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) ↔ (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
17	15, 16	bitr4di 198	. . . . . . . 8 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) ↔ ∃𝑞 ∈ Q ∃𝑟 ∈ Q ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))))
18	17	biimpa 296	. . . . . . 7 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → ∃𝑞 ∈ Q ∃𝑟 ∈ Q ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
19		simprll 537	. . . . . . . . . . 11 ⊢ ((((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 ∈ (2nd ‘𝑠))
20		prop 7685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
21		prltlu 7697	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
22	20, 21	syl3an1 1304	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
23	22	3adant3r 1259	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
24	23	3adant2l 1256	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ P ∧ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
25	24	3expb 1228	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
26	25	3ad2antl2 1184	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
27	26	adantlr 477	. . . . . . . . . . . 12 ⊢ ((((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
28		prop 7685	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ P → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P)
29		prcdnql 7694	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
30	28, 29	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
31	30	adantrl 478	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ P ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
32	31	adantrl 478	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
33	32	3ad2antl3 1185	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
34	33	adantlr 477	. . . . . . . . . . . 12 ⊢ ((((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
35	27, 34	mpd 13	. . . . . . . . . . 11 ⊢ ((((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 ∈ (1st ‘𝑢))
36	19, 35	jca 306	. . . . . . . . . 10 ⊢ ((((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢)))
37	36	ex 115	. . . . . . . . 9 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → (((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
38	37	rexlimdvw 2652	. . . . . . . 8 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → (∃𝑟 ∈ Q ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
39	38	reximdv 2631	. . . . . . 7 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → (∃𝑞 ∈ Q ∃𝑟 ∈ Q ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
40	18, 39	mpd 13	. . . . . 6 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢)))
41		ltdfpr 7716	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
42	41	3adant2 1040	. . . . . . . 8 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
43	42	biimprd 158	. . . . . . 7 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢)) → 𝑠<P 𝑢))
44	43	adantr 276	. . . . . 6 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢)) → 𝑠<P 𝑢))
45	40, 44	mpd 13	. . . . 5 ⊢ (((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑠<P 𝑡 ∧ 𝑡<P 𝑢)) → 𝑠<P 𝑢)
46	45	ex 115	. . . 4 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) → 𝑠<P 𝑢))
47	46	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P)) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) → 𝑠<P 𝑢))
48	10, 47	ispod 4399	. 2 ⊢ (⊤ → <P Po P)
49	48	mptru 1404	1 ⊢ <P Po P

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002  ⊤wtru 1396   ∈ wcel 2200  ∃wrex 2509  ⟨cop 3670   class class class wbr 4086   Po wpo 4389  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Qcnq 7490   <_Q cltq 7495  Pcnp 7501  <_P cltp 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-enq 7557  df-nqqs 7558  df-ltnqqs 7563  df-inp 7676  df-iltp 7680

This theorem is referenced by:  ltsopr  7806