Theorem ltexprlemrl 7418

Description: Lemma for ltexpri 7421. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩

Assertion

Ref	Expression
ltexprlemrl	⊢ (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemrl

Dummy variables 𝑧 𝑤 𝑢 𝑣 𝑓 𝑔 ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 7313	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4591	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 113	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 7283	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnmaddl 7298	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (1st ‘𝐵)) → ∃𝑣 ∈ Q (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))
7	5, 6	sylan 281	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) → ∃𝑣 ∈ Q (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))
8	2	simpld 111	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
9		prop 7283	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
10	8, 9	syl 14	. . . . . . 7 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
11		prarloc 7311	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
12	10, 11	sylan 281	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
13	12	ad2ant2r 500	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
14		simplll 522	. . . . . . . . . . 11 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝐴<P 𝐵)
15	14	adantr 274	. . . . . . . . . 10 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
16		simplrl 524	. . . . . . . . . 10 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st ‘𝐴))
17		elprnql 7289	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
18	10, 17	sylan 281	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
19	15, 16, 18	syl2anc 408	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ Q)
20		elprnql 7289	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (1st ‘𝐵)) → 𝑤 ∈ Q)
21	5, 20	sylan 281	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) → 𝑤 ∈ Q)
22	21	ad3antrrr 483	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤 ∈ Q)
23		nqtri3or 7204	. . . . . . . . 9 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 <Q 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 <Q 𝑧))
24	19, 22, 23	syl2anc 408	. . . . . . . 8 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 <Q 𝑧))
25		ltexnqq 7216	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 <Q 𝑤 ↔ ∃𝑠 ∈ Q (𝑧 +Q 𝑠) = 𝑤))
26	19, 22, 25	syl2anc 408	. . . . . . . . . . . 12 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤 ↔ ∃𝑠 ∈ Q (𝑧 +Q 𝑠) = 𝑤))
27	26	biimpa 294	. . . . . . . . . . 11 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) → ∃𝑠 ∈ Q (𝑧 +Q 𝑠) = 𝑤)
28		simprr 521	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑠) = 𝑤)
29	16	ad2antrr 479	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑧 ∈ (1st ‘𝐴))
30		simprl 520	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑠 ∈ Q)
31		simpr 109	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 <Q (𝑧 +Q 𝑣))
32		simplrr 525	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd ‘𝐴))
33		prcunqu 7293	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
34	10, 33	sylan 281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
35	15, 32, 34	syl2anc 408	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
36	31, 35	mpd 13	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴))
37	36	ad2antrr 479	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴))
38	19	ad2antrr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑧 ∈ Q)
39		simplrl 524	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
40	39	ad3antrrr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑣 ∈ Q)
41		addcomnqg 7189	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
42	41	adantl 275	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43		addassnqg 7190	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
44	43	adantl 275	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
45	38, 40, 30, 42, 44	caov32d 5951	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑣) +Q 𝑠) = ((𝑧 +Q 𝑠) +Q 𝑣))
46		simplrr 525	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))
47	46	ad3antrrr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))
48		oveq1 5781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 +Q 𝑠) = 𝑤 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑤 +Q 𝑣))
49	48	eleq1d 2208	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 +Q 𝑠) = 𝑤 → (((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st ‘𝐵) ↔ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵)))
50	28, 49	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st ‘𝐵) ↔ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵)))
51	47, 50	mpbird 166	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st ‘𝐵))
52	45, 51	eqeltrd 2216	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st ‘𝐵))
53		eleq1 2202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑧 +Q 𝑣) → (𝑦 ∈ (2nd ‘𝐴) ↔ (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
54		oveq1 5781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑠) = ((𝑧 +Q 𝑣) +Q 𝑠))
55	54	eleq1d 2208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑠) ∈ (1st ‘𝐵) ↔ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st ‘𝐵)))
56	53, 55	anbi12d 464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st ‘𝐵)) ↔ ((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st ‘𝐵))))
57	56	spcegv 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) → (((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st ‘𝐵)) → ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st ‘𝐵))))
58	57	anabsi5 568	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st ‘𝐵)) → ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st ‘𝐵)))
59	37, 52, 58	syl2anc 408	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st ‘𝐵)))
60		ltexprlem.1	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
61	60	ltexprlemell 7406	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (1st ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st ‘𝐵))))
62	30, 59, 61	sylanbrc 413	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑠 ∈ (1st ‘𝐶))
63	15, 8	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴 ∈ P)
64	63	ad2antrr 479	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝐴 ∈ P)
65	60	ltexprlempr 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
66	15, 65	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐶 ∈ P)
67	66	ad2antrr 479	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝐶 ∈ P)
68		df-iplp 7276	. . . . . . . . . . . . . . 15 ⊢ +P = (𝑥 ∈ P, 𝑤 ∈ P ↦ ⟨{𝑧 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑣 ∈ Q (𝑓 ∈ (1st ‘𝑥) ∧ 𝑣 ∈ (1st ‘𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}, {𝑧 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑣 ∈ Q (𝑓 ∈ (2nd ‘𝑥) ∧ 𝑣 ∈ (2nd ‘𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}⟩)
69		addclnq 7183	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
70	68, 69	genpprecll 7322	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑧 ∈ (1st ‘𝐴) ∧ 𝑠 ∈ (1st ‘𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
71	64, 67, 70	syl2anc 408	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 ∈ (1st ‘𝐴) ∧ 𝑠 ∈ (1st ‘𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
72	29, 62, 71	mp2and 429	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶)))
73	28, 72	eqeltrrd 2217	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
74	27, 73	rexlimddv 2554	. . . . . . . . . 10 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
75	74	ex 114	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤 → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
76	14	ad2antrr 479	. . . . . . . . . . 11 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝐴<P 𝐵)
77		simpr 109	. . . . . . . . . . . 12 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
78	16	adantr 274	. . . . . . . . . . . 12 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 ∈ (1st ‘𝐴))
79	77, 78	eqeltrrd 2217	. . . . . . . . . . 11 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑤 ∈ (1st ‘𝐴))
80		ltaddpr 7405	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴<P (𝐴 +P 𝐶))
81	8, 65, 80	syl2anc 408	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → 𝐴<P (𝐴 +P 𝐶))
82		ltprordil 7397	. . . . . . . . . . . . 13 ⊢ (𝐴<P (𝐴 +P 𝐶) → (1st ‘𝐴) ⊆ (1st ‘(𝐴 +P 𝐶)))
83	82	sseld 3096	. . . . . . . . . . . 12 ⊢ (𝐴<P (𝐴 +P 𝐶) → (𝑤 ∈ (1st ‘𝐴) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
84	81, 83	syl 14	. . . . . . . . . . 11 ⊢ (𝐴<P 𝐵 → (𝑤 ∈ (1st ‘𝐴) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
85	76, 79, 84	sylc 62	. . . . . . . . . 10 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
86	85	ex 114	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 = 𝑤 → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
87		prcdnql 7292	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → (𝑤 <Q 𝑧 → 𝑤 ∈ (1st ‘𝐴)))
88	10, 87	sylan 281	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → (𝑤 <Q 𝑧 → 𝑤 ∈ (1st ‘𝐴)))
89	15, 16, 88	syl2anc 408	. . . . . . . . . 10 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑤 <Q 𝑧 → 𝑤 ∈ (1st ‘𝐴)))
90	15, 89, 84	sylsyld 58	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑤 <Q 𝑧 → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
91	75, 86, 90	3jaod 1282	. . . . . . . 8 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ((𝑧 <Q 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 <Q 𝑧) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
92	24, 91	mpd 13	. . . . . . 7 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
93	92	ex 114	. . . . . 6 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → (𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
94	93	rexlimdvva 2557	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) → (∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
95	13, 94	mpd 13	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) ∧ (𝑣 ∈ Q ∧ (𝑤 +Q 𝑣) ∈ (1st ‘𝐵))) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
96	7, 95	rexlimddv 2554	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (1st ‘𝐵)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
97	96	ex 114	. 2 ⊢ (𝐴<P 𝐵 → (𝑤 ∈ (1st ‘𝐵) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
98	97	ssrdv 3103	1 ⊢ (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 961   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   +_Q cplq 7090   <_Q cltq 7093  Pcnp 7099   +_P cpp 7101  <_P cltp 7103

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  ltexpri  7421