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Theorem ltexprlemfl 7417

Description: Lemma for ltexpri 7421. 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 7313	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4591	. . . . 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 7416	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7276	. . . . 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 7183	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 7320	. . . 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 408	. . 3 ⊢ (𝐴<_P 𝐵 → (𝑧 ∈ (1^st ‘(𝐴 +_P 𝐶)) ↔ ∃𝑤 ∈ (1^st ‘𝐴)∃𝑢 ∈ (1^st ‘𝐶)𝑧 = (𝑤 +_Q 𝑢)))
10		simprr 521	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 = (𝑤 +_Q 𝑢))
11	4	ltexprlemell 7406	. . . . . . . . . . 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 480	. . . . . . . . 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 7283	. . . . . . . . . . . . . 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 7295	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
19	17, 18	syl3an1 1249	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
20	19	3adant2r 1211	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
21	20	3adant2r 1211	. . . . . . . . . 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 1213	. . . . . . . . 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 7208	. . . . . . . . . . . 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 7173	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4591	. . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . 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 7289	. . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . 13 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → 𝑢 ∈ Q)
35	34	adantrr 470	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑢 ∈ Q)
36	35	3adant3 1001	. . . . . . . . . . 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 7189	. . . . . . . . . . . 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 5940	. . . . . . . . . 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 7283	. . . . . . . . . . . . . 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 7292	. . . . . . . . . . . . 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 469	. . . . . . . . . . 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 1000	. . . . . . . . . 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 1181	. . . . . . 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 1870	. . . . . 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 2216	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 ∈ (1^st ‘𝐵))
52	51	expr 372	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶))) → (𝑧 = (𝑤 +_Q 𝑢) → 𝑧 ∈ (1^st ‘𝐵)))
53	52	rexlimdvva 2557	. . 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 3103	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 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

W3C validator