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Theorem ltexprlemfl 6583

Description: Lemma for ltexpri 6587. One directon 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 6488	. . . . . 6 ⊢ <_P ⊆ (P × P)
2	1	brel 4335	. . . . 5 ⊢ (A<_P B → (A ∈ P ∧ B ∈ P))
3	2	simpld 105	. . . 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 6582	. . . 4 ⊢ (A<_P B → 𝐶 ∈ P)
6		df-iplp 6451	. . . . 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 6359	. . . . 5 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g +_Q ℎ) ∈ Q)
8	6, 7	genpelvl 6495	. . . 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 391	. . 3 ⊢ (A<_P B → (z ∈ (1^st ‘(A +_P 𝐶)) ↔ ∃w ∈ (1^st ‘A)∃u ∈ (1^st ‘𝐶)z = (w +_Q u)))
10		simprr 484	. . . . . 6 ⊢ ((A<_P B ∧ ((w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶)) ∧ z = (w +_Q u))) → z = (w +_Q u))
11	4	ltexprlemell 6572	. . . . . . . . . . 11 ⊢ (u ∈ (1^st ‘𝐶) ↔ (u ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q u) ∈ (1^st ‘B))))
12	11	biimpi 113	. . . . . . . . . 10 ⊢ (u ∈ (1^st ‘𝐶) → (u ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q u) ∈ (1^st ‘B))))
13	12	ad2antlr 458	. . . . . . . . 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 262	. . . . . . . 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 107	. . . . . . 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 6458	. . . . . . . . . . . . . 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 6470	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ w ∈ (1^st ‘A) ∧ y ∈ (2^nd ‘A)) → w <_Q y)
19	17, 18	syl3an1 1167	. . . . . . . . . . . 12 ⊢ ((A<_P B ∧ w ∈ (1^st ‘A) ∧ y ∈ (2^nd ‘A)) → w <_Q y)
20	19	3adant2r 1129	. . . . . . . . . . 11 ⊢ ((A<_P B ∧ (w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶)) ∧ y ∈ (2^nd ‘A)) → w <_Q y)
21	20	3adant2r 1129	. . . . . . . . . 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 1131	. . . . . . . . 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 6384	. . . . . . . . . . . 12 ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → (f <_Q g ↔ (ℎ +_Q f) <_Q (ℎ +_Q g)))
24	23	adantl 262	. . . . . . . . . . 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 6349	. . . . . . . . . . . . . 14 ⊢ <_Q ⊆ (Q × Q)
26	25	brel 4335	. . . . . . . . . . . . 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 105	. . . . . . . . . . 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 107	. . . . . . . . . . 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 6458	. . . . . . . . . . . . . . . 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 6464	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ u ∈ (1^st ‘𝐶)) → u ∈ Q)
33	31, 32	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((A<_P B ∧ u ∈ (1^st ‘𝐶)) → u ∈ Q)
34	33	adantrl 447	. . . . . . . . . . . . 13 ⊢ ((A<_P B ∧ (w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶))) → u ∈ Q)
35	34	adantrr 448	. . . . . . . . . . . 12 ⊢ ((A<_P B ∧ ((w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶)) ∧ z = (w +_Q u))) → u ∈ Q)
36	35	3adant3 923	. . . . . . . . . . 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 6365	. . . . . . . . . . . 12 ⊢ ((f ∈ Q ∧ g ∈ Q) → (f +_Q g) = (g +_Q f))
38	37	adantl 262	. . . . . . . . . . 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 5612	. . . . . . . . . 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 107	. . . . . . . . . . . . . 14 ⊢ (A<_P B → B ∈ P)
41		prop 6458	. . . . . . . . . . . . . 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 6467	. . . . . . . . . . . . 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 267	. . . . . . . . . . . 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 447	. . . . . . . . . . 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 922	. . . . . . . . . 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 139	. . . . . . . . 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 1103	. . . . . . 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 1775	. . . . . 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 2111	. . . . 5 ⊢ ((A<_P B ∧ ((w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶)) ∧ z = (w +_Q u))) → z ∈ (1^st ‘B))
52	51	expr 357	. . . 4 ⊢ ((A<_P B ∧ (w ∈ (1^st ‘A) ∧ u ∈ (1^st ‘𝐶))) → (z = (w +_Q u) → z ∈ (1^st ‘B)))
53	52	rexlimdvva 2434	. . 3 ⊢ (A<_P B → (∃w ∈ (1^st ‘A)∃u ∈ (1^st ‘𝐶)z = (w +_Q u) → z ∈ (1^st ‘B)))
54	9, 53	sylbid 139	. 2 ⊢ (A<_P B → (z ∈ (1^st ‘(A +_P 𝐶)) → z ∈ (1^st ‘B)))
55	54	ssrdv 2945	1 ⊢ (A<_P B → (1^st ‘(A +_P 𝐶)) ⊆ (1^st ‘B))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 {crab 2304 ⊆ wss 2911 ⟨cop 3370 class class class wbr 3755 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 +_Q cplq 6266 <_Q cltq 6269 Pcnp 6275 +_P cpp 6277 <_P cltp 6279

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-iplp 6451 df-iltp 6453

This theorem is referenced by: ltexpri 6587

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