Theorem ltpopr 7608

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 7609. (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 7488	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
2		prdisj 7505	. . . . . . . 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 677	. . . . . 6 ⊢ ((𝑠 ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
6	5	nrexdv 2580	. . . . 5 ⊢ (𝑠 ∈ P → ¬ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑠)))
7		ltdfpr 7519	. . . . . 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 674	. . . 4 ⊢ (𝑠 ∈ P → ¬ 𝑠<P 𝑠)
10	9	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑠 ∈ P) → ¬ 𝑠<P 𝑠)
11		ltdfpr 7519	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
12	11	3adant3 1018	. . . . . . . . . 10 ⊢ ((𝑠 ∈ P ∧ 𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑡 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡))))
13		ltdfpr 7519	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ P ∧ 𝑢 ∈ P) → (𝑡<P 𝑢 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))))
14	13	3adant1 1016	. . . . . . . . . 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 2657	. . . . . . . . 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 7488	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
21		prltlu 7500	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
22	20, 21	syl3an1 1281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ 𝑟 ∈ (2nd ‘𝑡)) → 𝑞 <Q 𝑟)
23	22	3adant3r 1236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ P ∧ 𝑞 ∈ (1st ‘𝑡) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
24	23	3adant2l 1233	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ P ∧ (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢))) → 𝑞 <Q 𝑟)
25	24	3expb 1205	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ P ∧ ((𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑡)) ∧ (𝑟 ∈ (2nd ‘𝑡) ∧ 𝑟 ∈ (1st ‘𝑢)))) → 𝑞 <Q 𝑟)
26	25	3ad2antl2 1161	. . . . . . . . . . . . 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 7488	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ P → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ P)
29		prcdnql 7497	. . . . . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . 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 2608	. . . . . . . 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 2588	. . . . . . 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 7519	. . . . . . . . 9 ⊢ ((𝑠 ∈ P ∧ 𝑢 ∈ P) → (𝑠<P 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑠) ∧ 𝑞 ∈ (1st ‘𝑢))))
42	41	3adant2 1017	. . . . . . . 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 4316	. 2 ⊢ (⊤ → <P Po P)
49	48	mptru 1372	1 ⊢ <P Po P

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 979  ⊤wtru 1364   ∈ wcel 2158  ∃wrex 2466  ⟨cop 3607   class class class wbr 4015   Po wpo 4306  ‘cfv 5228  1^st c1st 6153  2^nd c2nd 6154  Qcnq 7293   <_Q cltq 7298  Pcnp 7304  <_P cltp 7308

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-mi 7319  df-lti 7320  df-enq 7360  df-nqqs 7361  df-ltnqqs 7366  df-inp 7479  df-iltp 7483

This theorem is referenced by:  ltsopr  7609