ltaddpr - Intuitionistic Logic Explorer

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Theorem ltaddpr 6571

Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)

**Assertion**
Ref	Expression
ltaddpr	⊢ ((A ∈ P ∧ B ∈ P) → A<_P (A +_P B))

Proof of Theorem ltaddpr Dummy variables f g ℎ x y 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6458	. . . 4 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
2		prml 6460	. . . 4 ⊢ (⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘B))
3	1, 2	syl 14	. . 3 ⊢ (B ∈ P → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘B))
4	3	adantl 262	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘B))
5		prop 6458	. . . . 5 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
6		prarloc 6486	. . . . 5 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ 𝑝 ∈ Q) → ∃𝑟 ∈ (1^st ‘A)∃𝑞 ∈ (2^nd ‘A)𝑞 <_Q (𝑟 +_Q 𝑝))
7	5, 6	sylan 267	. . . 4 ⊢ ((A ∈ P ∧ 𝑝 ∈ Q) → ∃𝑟 ∈ (1^st ‘A)∃𝑞 ∈ (2^nd ‘A)𝑞 <_Q (𝑟 +_Q 𝑝))
8	7	ad2ant2r 478	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) → ∃𝑟 ∈ (1^st ‘A)∃𝑞 ∈ (2^nd ‘A)𝑞 <_Q (𝑟 +_Q 𝑝))
9		elprnqu 6465	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ 𝑞 ∈ (2^nd ‘A)) → 𝑞 ∈ Q)
10	5, 9	sylan 267	. . . . . . . . . 10 ⊢ ((A ∈ P ∧ 𝑞 ∈ (2^nd ‘A)) → 𝑞 ∈ Q)
11	10	adantlr 446	. . . . . . . . 9 ⊢ (((A ∈ P ∧ B ∈ P) ∧ 𝑞 ∈ (2^nd ‘A)) → 𝑞 ∈ Q)
12	11	ad2ant2rl 480	. . . . . . . 8 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → 𝑞 ∈ Q)
13	12	adantr 261	. . . . . . 7 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ Q)
14		simplrr 488	. . . . . . 7 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ (2^nd ‘A))
15		simprl 483	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → 𝑟 ∈ (1^st ‘A))
16		simplr 482	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → 𝑝 ∈ (1^st ‘B))
17	15, 16	jca 290	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → (𝑟 ∈ (1^st ‘A) ∧ 𝑝 ∈ (1^st ‘B)))
18		df-iplp 6451	. . . . . . . . . . . . 13 ⊢ +_P = (x ∈ P, y ∈ P ↦ ⟨{f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (1^st ‘x) ∧ ℎ ∈ (1^st ‘y) ∧ f = (g +_Q ℎ))}, {f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (2^nd ‘x) ∧ ℎ ∈ (2^nd ‘y) ∧ f = (g +_Q ℎ))}⟩)
19		addclnq 6359	. . . . . . . . . . . . 13 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g +_Q ℎ) ∈ Q)
20	18, 19	genpprecll 6497	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ B ∈ P) → ((𝑟 ∈ (1^st ‘A) ∧ 𝑝 ∈ (1^st ‘B)) → (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))))
21	17, 20	syl5 28	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))))
22	21	imdistani 419	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ ((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A)))) → ((A ∈ P ∧ B ∈ P) ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))))
23		addclpr 6520	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → (A +_P B) ∈ P)
24		prop 6458	. . . . . . . . . . . 12 ⊢ ((A +_P B) ∈ P → ⟨(1^st ‘(A +_P B)), (2^nd ‘(A +_P B))⟩ ∈ P)
25		prcdnql 6467	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘(A +_P B)), (2^nd ‘(A +_P B))⟩ ∈ P ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(A +_P B))))
26	24, 25	sylan 267	. . . . . . . . . . 11 ⊢ (((A +_P B) ∈ P ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(A +_P B))))
27	23, 26	sylan 267	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(A +_P B))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(A +_P B))))
28	22, 27	syl 14	. . . . . . . . 9 ⊢ (((A ∈ P ∧ B ∈ P) ∧ ((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B)) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A)))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(A +_P B))))
29	28	anassrs 380	. . . . . . . 8 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(A +_P B))))
30	29	imp 115	. . . . . . 7 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ (1^st ‘(A +_P B)))
31		rspe 2364	. . . . . . 7 ⊢ ((𝑞 ∈ Q ∧ (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B)))) → ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B))))
32	13, 14, 30, 31	syl12anc 1132	. . . . . 6 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B))))
33		ltdfpr 6489	. . . . . . . 8 ⊢ ((A ∈ P ∧ (A +_P B) ∈ P) → (A<_P (A +_P B) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B)))))
34	23, 33	syldan 266	. . . . . . 7 ⊢ ((A ∈ P ∧ B ∈ P) → (A<_P (A +_P B) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B)))))
35	34	ad3antrrr 461	. . . . . 6 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → (A<_P (A +_P B) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘A) ∧ 𝑞 ∈ (1^st ‘(A +_P B)))))
36	32, 35	mpbird 156	. . . . 5 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → A<_P (A +_P B))
37	36	ex 108	. . . 4 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) ∧ (𝑟 ∈ (1^st ‘A) ∧ 𝑞 ∈ (2^nd ‘A))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → A<_P (A +_P B)))
38	37	rexlimdvva 2434	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) → (∃𝑟 ∈ (1^st ‘A)∃𝑞 ∈ (2^nd ‘A)𝑞 <_Q (𝑟 +_Q 𝑝) → A<_P (A +_P B)))
39	8, 38	mpd 13	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘B))) → A<_P (A +_P B))
40	4, 39	rexlimddv 2431	1 ⊢ ((A ∈ P ∧ B ∈ P) → A<_P (A +_P B))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∃wrex 2301 ⟨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: ltexprlemrl 6584 ltaprlem 6591 ltaprg 6592 ltmprr 6614 0lt1sr 6693 recexgt0sr 6701 mulgt0sr 6704 archsr 6708

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