Theorem ltpopr 7304

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 7305. (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 7184	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
2		prdisj 7201	. . . . . . . 8 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
3	1, 2	sylan 279	. . . . . . 7 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
4		ancom 264	. . . . . . 7 ⊢ ((𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)) ↔ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
5	3, 4	sylnib 642	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
6	5	nrexdv 2484	. . . . 5 ⊢ (𝑠 ∈ P → ¬ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
7		ltdfpr 7215	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑠 ∈ P) → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
8	7	anidms 392	. . . . 5 ⊢ (𝑠 ∈ P → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
9	6, 8	mtbird 639	. . . 4 ⊢ (𝑠 ∈ P → ¬ 𝑠<P 𝑠)
10	9	adantl 273	. . 3 ⊢ ((⊤ ∧ 𝑠 ∈ P) → ¬ 𝑠<P 𝑠)
11		ltdfpr 7215	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
12	11	3adant3 969	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
13		ltdfpr 7215	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
14	13	3adant1 967	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
15	12, 14	anbi12d 460	. . . . . . . . 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 2558	. . . . . . . . 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	syl6bbr 197	. . . . . . . 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 292	. . . . . . 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 507	. . . . . . . . . . 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 7184	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
21		prltlu 7196	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
22	20, 21	syl3an1 1217	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
23	22	3adant3r 1181	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
24	23	3adant2l 1178	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ P ∧ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
25	24	3expb 1150	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
26	25	3ad2antl2 1112	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 7184	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ P → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P)
29		prcdnql 7193	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
30	28, 29	sylan 279	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
31	30	adantrl 465	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ P ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
32	31	adantrl 465	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
33	32	3ad2antl3 1113	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 302	. . . . . . . . . 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 114	. . . . . . . . 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 2512	. . . . . . . 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 2492	. . . . . . 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 7215	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
42	41	3adant2 968	. . . . . . . 8 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
43	42	biimprd 157	. . . . . . 7 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢)) → 𝑠<P 𝑢))
44	43	adantr 272	. . . . . 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 114	. . . 4 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) → 𝑠<P 𝑢))
47	46	adantl 273	. . 3 ⊢ ((⊤ ∧ (𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P)) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) → 𝑠<P 𝑢))
48	10, 47	ispod 4164	. 2 ⊢ (⊤ → <P Po P)
49	48	mptru 1308	1 ⊢ <P Po P

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 930  ⊤wtru 1300   ∈ wcel 1448  ∃wrex 2376  ⟨cop 3477   class class class wbr 3875   Po wpo 4154  ‘cfv 5059  1^st c1st 5967  2^nd c2nd 5968  Qcnq 6989   <_Q cltq 6994  Pcnp 7000  <_P cltp 7004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-mi 7015  df-lti 7016  df-enq 7056  df-nqqs 7057  df-ltnqqs 7062  df-inp 7175  df-iltp 7179

This theorem is referenced by:  ltsopr  7305