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Theorem ltexprlemfl 7441

Description: Lemma for ltexpri 7445. 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 7337	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4599	. . . . 5 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simpld 111	. . . 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 7440	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7300	. . . . 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 7207	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 7344	. . . 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 409	. . 3 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) ↔ ∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢)))
10		simprr 522	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 = (𝑤 +_Q 𝑢))
11	4	ltexprlemell 7430	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (1^st ‘𝐶) ↔ (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
12	11	biimpi 119	. . . . . . . . . 10 ⊢ (𝑢 ∈ (1^st ‘𝐶) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑢) ∈ (1^st ‘𝐵))))
13	12	ad2antlr 481	. . . . . . . . 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 275	. . . . . . . 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 113	. . . . . . 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 7307	. . . . . . . . . . . . . 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 7319	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
19	17, 18	syl3an1 1250	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
20	19	3adant2r 1212	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
21	20	3adant2r 1212	. . . . . . . . . 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 1214	. . . . . . . . 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 7232	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
24	23	adantl 275	. . . . . . . . . . 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 7197	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4599	. . . . . . . . . . . . 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 111	. . . . . . . . . . 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 113	. . . . . . . . . . 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 7307	. . . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑢 ∈ (1^st ‘𝐶)) → 𝑢 ∈ Q)
33	31, 32	sylan 281	. . . . . . . . . . . . . 14 ⊢ ((𝐴<_P 𝐵 ∧ 𝑢 ∈ (1^st ‘𝐶)) → 𝑢 ∈ Q)
34	33	adantrl 470	. . . . . . . . . . . . 13 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → 𝑢 ∈ Q)
35	34	adantrr 471	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑢 ∈ Q)
36	35	3adant3 1002	. . . . . . . . . . 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 7213	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
38	37	adantl 275	. . . . . . . . . . 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 5948	. . . . . . . . . 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 113	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
41		prop 7307	. . . . . . . . . . . . . 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 7316	. . . . . . . . . . . . 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 281	. . . . . . . . . . . 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 470	. . . . . . . . . . 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 1001	. . . . . . . . . 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 149	. . . . . . . . 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 1182	. . . . . . 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 1871	. . . . . 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 2217	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 ∈ (1^st ‘𝐵))
52	51	expr 373	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → (𝑧 = (𝑤 +_Q 𝑢) → 𝑧 ∈ (1^st ‘𝐵)))
53	52	rexlimdvva 2560	. . 3 ⊢ (𝐴<_P 𝐵 → (∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢) → 𝑧 ∈ (1^st ‘𝐵)))
54	9, 53	sylbid 149	. 2 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) → 𝑧 ∈ (1^st ‘𝐵)))
55	54	ssrdv 3108	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P 𝐶)) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∃wrex 2418 {crab 2421 ⊆ wss 3076 ⟨cop 3535 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 1^st c1st 6044 2^nd c2nd 6045 Qcnq 7112 +_Q cplq 7114 <_Q cltq 7117 Pcnp 7123 +_P cpp 7125 <_P cltp 7127

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-iltp 7302

This theorem is referenced by: ltexpri 7445

W3C validator