supxrun - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > supxrun		Structured version Visualization version GIF version

Theorem supxrun 13358

Description: The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
supxrun	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup((𝐴 ∪ 𝐵), ℝ^, < ) = sup(𝐵, ℝ^, < ))

Proof of Theorem supxrun Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unss 4190	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ^)
2	1	biimpi 216	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐴 ∪ 𝐵) ⊆ ℝ^)
3	2	3adant3 1133	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝐴 ∪ 𝐵) ⊆ ℝ^*)
4		supxrcl 13357	. . 3 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
5	4	3ad2ant2 1135	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup(𝐵, ℝ^, < ) ∈ ℝ^)
6		elun 4153	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
7		xrltso 13183	. . . . . . . . 9 ⊢ < Or ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
9		xrsupss 13351	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 < 𝑤)))
10	8, 9	supub 9499	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
11	10	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
12		supxrcl 13357	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
13	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
14	4	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐵, ℝ^, < ) ∈ ℝ^*)
15		ssel2 3978	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16	15	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
17		xrlelttr 13198	. . . . . . . . . . . 12 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐵, ℝ^, < ) ∈ ℝ^ ∧ 𝑥 ∈ ℝ^) → ((sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ∧ sup(𝐵, ℝ^, < ) < 𝑥) → sup(𝐴, ℝ^*, < ) < 𝑥))
18	13, 14, 16, 17	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → ((sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ∧ sup(𝐵, ℝ^, < ) < 𝑥) → sup(𝐴, ℝ^*, < ) < 𝑥))
19	18	expdimp 452	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (sup(𝐵, ℝ^, < ) < 𝑥 → sup(𝐴, ℝ^*, < ) < 𝑥))
20	19	con3d 152	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
21	20	exp41 434	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (𝐵 ⊆ ℝ^* → (𝑥 ∈ 𝐴 → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^, < ) < 𝑥)))))
22	21	com34 91	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝐵 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) → (𝑥 ∈ 𝐴 → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^, < ) < 𝑥)))))
23	22	3imp 1111	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐴 → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^, < ) < 𝑥)))
24	11, 23	mpdd 43	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐴 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
25	7	a1i 11	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → < Or ℝ^*)
26		xrsupss 13351	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐵 𝑧 < 𝑤)))
27	25, 26	supub 9499	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
28	27	3ad2ant2 1135	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
29	24, 28	jaod 860	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
30	6, 29	biimtrid 242	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ (𝐴 ∪ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
31	30	ralrimiv 3145	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^*, < ) < 𝑥)
32		rexr 11307	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
33		xrsupss 13351	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑥 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑥 → ∃𝑦 ∈ 𝐵 𝑧 < 𝑦)))
34	25, 33	suplub 9500	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
35	32, 34	sylani 604	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
36		elun2 4183	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
37	36	anim1i 615	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 < 𝑦))
38	37	reximi2 3079	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑥 < 𝑦 → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦)
39	35, 38	syl6 35	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
40	39	expd 415	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ ℝ → (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦)))
41	40	ralrimiv 3145	. . 3 ⊢ (𝐵 ⊆ ℝ^* → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
42	41	3ad2ant2 1135	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
43		supxr 13355	. 2 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ℝ^* ∧ sup(𝐵, ℝ^, < ) ∈ ℝ^) ∧ (∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^, < ) < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))) → sup((𝐴 ∪ 𝐵), ℝ^, < ) = sup(𝐵, ℝ^, < ))
44	3, 5, 31, 42, 43	syl22anc 839	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup((𝐴 ∪ 𝐵), ℝ^, < ) = sup(𝐵, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∪ cun 3949 ⊆ wss 3951 class class class wbr 5143 Or wor 5591 supcsup 9480 ℝcr 11154 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495

This theorem is referenced by: supxrmnf 13359 xpsdsval 24391

W3C validator