Theorem ltpopr 7396

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 7397. (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 7276	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
2		prdisj 7293	. . . . . . . 8 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
3	1, 2	sylan 281	. . . . . . 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 665	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
6	5	nrexdv 2523	. . . . 5 ⊢ (𝑠 ∈ P → ¬ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
7		ltdfpr 7307	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑠 ∈ P) → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
8	7	anidms 394	. . . . 5 ⊢ (𝑠 ∈ P → (𝑠<P 𝑠 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠))))
9	6, 8	mtbird 662	. . . 4 ⊢ (𝑠 ∈ P → ¬ 𝑠<P 𝑠)
10	9	adantl 275	. . 3 ⊢ ((⊤ ∧ 𝑠 ∈ P) → ¬ 𝑠<P 𝑠)
11		ltdfpr 7307	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
12	11	3adant3 1001	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
13		ltdfpr 7307	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
14	13	3adant1 999	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
15	12, 14	anbi12d 464	. . . . . . . . 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 2598	. . . . . . . . 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 294	. . . . . . 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 526	. . . . . . . . . . 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 7276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
21		prltlu 7288	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
22	20, 21	syl3an1 1249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
23	22	3adant3r 1213	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
24	23	3adant2l 1210	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ P ∧ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
25	24	3expb 1182	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
26	25	3ad2antl2 1144	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 7276	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ P → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P)
29		prcdnql 7285	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
30	28, 29	sylan 281	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ P ∧ 𝑟 ∈ (1st ‘𝑢)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
31	30	adantrl 469	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ P ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
32	31	adantrl 469	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑢)))
33	32	3ad2antl3 1145	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 304	. . . . . . . . . 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 2551	. . . . . . . 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 2531	. . . . . . 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 7307	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
42	41	3adant2 1000	. . . . . . . 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 274	. . . . . 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 275	. . 3 ⊢ ((⊤ ∧ (𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P)) → ((𝑠<P 𝑡 ∧ 𝑡<P 𝑢) → 𝑠<P 𝑢))
48	10, 47	ispod 4221	. 2 ⊢ (⊤ → <P Po P)
49	48	mptru 1340	1 ⊢ <P Po P

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962  ⊤wtru 1332   ∈ wcel 1480  ∃wrex 2415  ⟨cop 3525   class class class wbr 3924   Po wpo 4211  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081   <_Q cltq 7086  Pcnp 7092  <_P cltp 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-enq 7148  df-nqqs 7149  df-ltnqqs 7154  df-inp 7267  df-iltp 7271

This theorem is referenced by:  ltsopr  7397