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Theorem ltexprlemfl 7982

Description: Lemma for ltexpri 7986. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

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

**Assertion**
Ref	Expression
ltexprlemfl	⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P 𝐶)) ⊆ (1^st ‘𝐵))

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

Proof of Theorem ltexprlemfl Dummy variables 𝑧 𝑤 𝑢 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 7878	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4829	. . . . 5 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simpld 112	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
4		ltexprlem.1	. . . . 5 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
5	4	ltexprlempr 7981	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7841	. . . . 5 ⊢ +_P = (𝑧 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑧) ∧ ℎ ∈ (1^st ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑧) ∧ ℎ ∈ (2^nd ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}⟩)
7		addclnq 7748	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 7885	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) ↔ ∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢)))
9	3, 5, 8	syl2anc 415	. . 3 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) ↔ ∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢)))
10		simprr 537	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 = (𝑤 +_Q 𝑢))
11	4	ltexprlemell 7971	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (1^st ‘𝐶) ↔ (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
12	11	biimpi 120	. . . . . . . . . 10 ⊢ (𝑢 ∈ (1^st ‘𝐶) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
13	12	ad2antlr 493	. . . . . . . . 9 ⊢ (((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
14	13	adantl 277	. . . . . . . 8 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
15	14	simprd 114	. . . . . . 7 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵)))
16		prop 7848	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
17	3, 16	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
18		prltlu 7860	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
19	17, 18	syl3an1 1311	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
20	19	3adant2r 1264	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
21	20	3adant2r 1264	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
22	21	3adant3r 1266	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → 𝑤 <_Q 𝑦)
23		ltanqg 7773	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
24	23	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
25		ltrelnq 7738	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4829	. . . . . . . . . . . . 13 ⊢ (𝑤 <_Q 𝑦 → (𝑤 ∈ Q ∧ 𝑦 ∈ Q))
27	22, 26	syl 14	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → (𝑤 ∈ Q ∧ 𝑦 ∈ Q))
28	27	simpld 112	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → 𝑤 ∈ Q)
29	27	simprd 114	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → 𝑦 ∈ Q)
30		prop 7848	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
31	5, 30	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
32		elprnql 7854	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑢 ∈ (1^st ‘𝐶)) → 𝑢 ∈ Q)
33	31, 32	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴<_P 𝐵 ∧ 𝑢 ∈ (1^st ‘𝐶)) → 𝑢 ∈ Q)
34	33	adantrl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → 𝑢 ∈ Q)
35	34	adantrr 483	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑢 ∈ Q)
36	35	3adant3 1048	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → 𝑢 ∈ Q)
37		addcomnqg 7754	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
38	37	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
39	24, 28, 29, 36, 38	caovord2d 6262	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → (𝑤 <_Q 𝑦 ↔ (𝑤 +_Q 𝑢) <_Q (𝑦 +_Q 𝑢)))
40	2	simprd 114	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
41		prop 7848	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
42	40, 41	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
43		prcdnql 7857	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵)) → ((𝑤 +_Q 𝑢) <_Q (𝑦 +_Q 𝑢) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵)))
44	42, 43	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵)) → ((𝑤 +_Q 𝑢) <_Q (𝑦 +_Q 𝑢) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵)))
45	44	adantrl 482	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → ((𝑤 +_Q 𝑢) <_Q (𝑦 +_Q 𝑢) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵)))
46	45	3adant2 1047	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → ((𝑤 +_Q 𝑢) <_Q (𝑦 +_Q 𝑢) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵)))
47	39, 46	sylbid 150	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → (𝑤 <_Q 𝑦 → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵)))
48	22, 47	mpd 13	. . . . . . . 8 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢)) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵))
49	48	3expa 1234	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵))
50	15, 49	exlimddv 1954	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → (𝑤 +_Q 𝑢) ∈ (1^st ‘𝐵))
51	10, 50	eqeltrd 2315	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 ∈ (1^st ‘𝐵))
52	51	expr 375	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → (𝑧 = (𝑤 +_Q 𝑢) → 𝑧 ∈ (1^st ‘𝐵)))
53	52	rexlimdvva 2676	. . 3 ⊢ (𝐴<_P 𝐵 → (∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢) → 𝑧 ∈ (1^st ‘𝐵)))
54	9, 53	sylbid 150	. 2 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) → 𝑧 ∈ (1^st ‘𝐵)))
55	54	ssrdv 3254	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P 𝐶)) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∃wrex 2529 {crab 2532 ⊆ wss 3220 ⟨cop 3712 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 Qcnq 7653 +_Q cplq 7655 <_Q cltq 7658 Pcnp 7664 +_P cpp 7666 <_P cltp 7668

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-1o 6691 df-2o 6692 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7677 df-pli 7678 df-mi 7679 df-lti 7680 df-plpq 7717 df-mpq 7718 df-enq 7720 df-nqqs 7721 df-plqqs 7722 df-mqqs 7723 df-1nqqs 7724 df-rq 7725 df-ltnqqs 7726 df-enq0 7797 df-nq0 7798 df-0nq0 7799 df-plq0 7800 df-mq0 7801 df-inp 7839 df-iplp 7841 df-iltp 7843

This theorem is used by: ltexpri 7986

W3C validator