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Theorem ltexprlemfl 7940

Description: Lemma for ltexpri 7944. 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 7836	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4807	. . . . 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 7939	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7799	. . . . 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 7706	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 7843	. . . 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 411	. . 3 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) ↔ ∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢)))
10		simprr 533	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 = (𝑤 +_Q 𝑢))
11	4	ltexprlemell 7929	. . . . . . . . . . 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 489	. . . . . . . . 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 7806	. . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
19	17, 18	syl3an1 1307	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
20	19	3adant2r 1260	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
21	20	3adant2r 1260	. . . . . . . . . 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 1262	. . . . . . . . 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 7731	. . . . . . . . . . . 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 7696	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4807	. . . . . . . . . . . . 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 7806	. . . . . . . . . . . . . . . 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 7812	. . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → 𝑢 ∈ Q)
35	34	adantrr 479	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑢 ∈ Q)
36	35	3adant3 1044	. . . . . . . . . . 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 7712	. . . . . . . . . . . 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 6232	. . . . . . . . . 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 7806	. . . . . . . . . . . . . 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 7815	. . . . . . . . . . . . 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 478	. . . . . . . . . . 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 1043	. . . . . . . . . 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 1230	. . . . . . 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 1950	. . . . . 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 2311	. . . . 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 2670	. . 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 3248	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P 𝐶)) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2205 ∃wrex 2523 {crab 2526 ⊆ wss 3214 ⟨cop 3697 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 1^st c1st 6345 2^nd c2nd 6346 Qcnq 7611 +_Q cplq 7613 <_Q cltq 7616 Pcnp 7622 +_P cpp 7624 <_P cltp 7626

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-eprel 4415 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-1o 6660 df-2o 6661 df-oadd 6664 df-omul 6665 df-er 6780 df-ec 6782 df-qs 6786 df-ni 7635 df-pli 7636 df-mi 7637 df-lti 7638 df-plpq 7675 df-mpq 7676 df-enq 7678 df-nqqs 7679 df-plqqs 7680 df-mqqs 7681 df-1nqqs 7682 df-rq 7683 df-ltnqqs 7684 df-enq0 7755 df-nq0 7756 df-0nq0 7757 df-plq0 7758 df-mq0 7759 df-inp 7797 df-iplp 7799 df-iltp 7801

This theorem is referenced by: ltexpri 7944

W3C validator