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Theorem ltexprlemfl 7599

Description: Lemma for ltexpri 7603. 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 7495	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4675	. . . . 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 7598	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7458	. . . . 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 7365	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 7502	. . . 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 531	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑧 = (𝑤 +_Q 𝑢))) → 𝑧 = (𝑤 +_Q 𝑢))
11	4	ltexprlemell 7588	. . . . . . . . . . 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 7465	. . . . . . . . . . . . . 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 7477	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
19	17, 18	syl3an1 1271	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑤 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
20	19	3adant2r 1233	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑤 ∈ (1^st ‘𝐴) ∧ 𝑢 ∈ (1^st ‘𝐶)) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑤 <_Q 𝑦)
21	20	3adant2r 1233	. . . . . . . . . 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 1235	. . . . . . . . 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 7390	. . . . . . . . . . . 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 7355	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4675	. . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . 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 7471	. . . . . . . . . . . . . . 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 1017	. . . . . . . . . . 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 7371	. . . . . . . . . . . 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 6038	. . . . . . . . . 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 7465	. . . . . . . . . . . . . 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 7474	. . . . . . . . . . . . 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 1016	. . . . . . . . . 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 1203	. . . . . . 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 1898	. . . . . 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 2254	. . . . 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 2602	. . 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 3161	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 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⊆ wss 3129 ⟨cop 3594 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 1^st c1st 6133 2^nd c2nd 6134 Qcnq 7270 +_Q cplq 7272 <_Q cltq 7275 Pcnp 7281 +_P cpp 7283 <_P cltp 7285

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7294 df-pli 7295 df-mi 7296 df-lti 7297 df-plpq 7334 df-mpq 7335 df-enq 7337 df-nqqs 7338 df-plqqs 7339 df-mqqs 7340 df-1nqqs 7341 df-rq 7342 df-ltnqqs 7343 df-enq0 7414 df-nq0 7415 df-0nq0 7416 df-plq0 7417 df-mq0 7418 df-inp 7456 df-iplp 7458 df-iltp 7460

This theorem is referenced by: ltexpri 7603

W3C validator