Theorem ltpopr 7596

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 7597. (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 7476	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
2		prdisj 7493	. . . . . . . 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 676	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
6	5	nrexdv 2570	. . . . 5 ⊢ (𝑠 ∈ P → ¬ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
7		ltdfpr 7507	. . . . . 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 673	. . . 4 ⊢ (𝑠 ∈ P → ¬ 𝑠<P 𝑠)
10	9	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑠 ∈ P) → ¬ 𝑠<P 𝑠)
11		ltdfpr 7507	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
12	11	3adant3 1017	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
13		ltdfpr 7507	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
14	13	3adant1 1015	. . . . . . . . . 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 2647	. . . . . . . . 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 7476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
21		prltlu 7488	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
22	20, 21	syl3an1 1271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
23	22	3adant3r 1235	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
24	23	3adant2l 1232	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ P ∧ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
25	24	3expb 1204	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
26	25	3ad2antl2 1160	. . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ P → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P)
29		prcdnql 7485	. . . . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . 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 2598	. . . . . . . 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 2578	. . . . . . 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 7507	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
42	41	3adant2 1016	. . . . . . . 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 4306	. 2 ⊢ (⊤ → <P Po P)
49	48	mptru 1362	1 ⊢ <P Po P

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  ⊤wtru 1354   ∈ wcel 2148  ∃wrex 2456  ⟨cop 3597   class class class wbr 4005   Po wpo 4296  ‘cfv 5218  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   <_Q cltq 7286  Pcnp 7292  <_P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-enq 7348  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  ltsopr  7597