Theorem addnqprlemfu 7573

Description: Lemma for addnqpr 7574. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemfu	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))

Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem addnqprlemfu

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addnqprlemrl 7570	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
2		ltsonq 7411	. . . . . . . . 9 ⊢ <Q Or Q
3		addclnq 7388	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
4		sonr 4329	. . . . . . . . 9 ⊢ (( <Q Or Q ∧ (𝐴 +Q 𝐵) ∈ Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
5	2, 3, 4	sylancr 414	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
6		ltrelnq 7378	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
7	6	brel 4690	. . . . . . . . . . 11 ⊢ ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → ((𝐴 +Q 𝐵) ∈ Q ∧ (𝐴 +Q 𝐵) ∈ Q))
8	7	simpld 112	. . . . . . . . . 10 ⊢ ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ Q)
9		elex 2760	. . . . . . . . . 10 ⊢ ((𝐴 +Q 𝐵) ∈ Q → (𝐴 +Q 𝐵) ∈ V)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ V)
11		breq1 4018	. . . . . . . . 9 ⊢ (𝑙 = (𝐴 +Q 𝐵) → (𝑙 <Q (𝐴 +Q 𝐵) ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵)))
12	10, 11	elab3 2901	. . . . . . . 8 ⊢ ((𝐴 +Q 𝐵) ∈ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)} ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
13	5, 12	sylnibr 678	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ¬ (𝐴 +Q 𝐵) ∈ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)})
14		ltnqex 7562	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
15		gtnqex 7563	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
16	14, 15	op1st 6161	. . . . . . . 8 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}
17	16	eleq2i 2254	. . . . . . 7 ⊢ ((𝐴 +Q 𝐵) ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) ∈ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)})
18	13, 17	sylnibr 678	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
19	1, 18	ssneldd 3170	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
20	19	adantr 276	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
21		nqprlu 7560	. . . . . . . 8 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
22		nqprlu 7560	. . . . . . . 8 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
23		addclpr 7550	. . . . . . . 8 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P)
24	21, 22, 23	syl2an 289	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P)
25		prop 7488	. . . . . . 7 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))⟩ ∈ P)
26	24, 25	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨(1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))⟩ ∈ P)
27		vex 2752	. . . . . . . 8 ⊢ 𝑟 ∈ V
28		breq2 4019	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((𝐴 +Q 𝐵) <Q 𝑢 ↔ (𝐴 +Q 𝐵) <Q 𝑟))
29	14, 15	op2nd 6162	. . . . . . . 8 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}
30	27, 28, 29	elab2 2897	. . . . . . 7 ⊢ (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) <Q 𝑟)
31	30	biimpi 120	. . . . . 6 ⊢ (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → (𝐴 +Q 𝐵) <Q 𝑟)
32		prloc 7504	. . . . . 6 ⊢ ((⟨(1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))⟩ ∈ P ∧ (𝐴 +Q 𝐵) <Q 𝑟) → ((𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
33	26, 31, 32	syl2an 289	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → ((𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
34	33	orcomd 730	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ∨ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
35	20, 34	ecased 1359	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
36	35	ex 115	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
37	36	ssrdv 3173	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   ∈ wcel 2158  {cab 2173  Vcvv 2749   ⊆ wss 3141  ⟨cop 3607   class class class wbr 4015   Or wor 4307  ‘cfv 5228  (class class class)co 5888  1^st c1st 6153  2^nd c2nd 6154  Qcnq 7293   +_Q cplq 7295   <_Q cltq 7298  Pcnp 7304   +_P cpp 7306

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-iplp 7481

This theorem is referenced by:  addnqpr  7574