supxrun - Metamath Proof Explorer

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Theorem supxrun 13332

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 4165	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ^)
2	1	biimpi 216	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐴 ∪ 𝐵) ⊆ ℝ^)
3	2	3adant3 1132	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝐴 ∪ 𝐵) ⊆ ℝ^*)
4		supxrcl 13331	. . 3 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
5	4	3ad2ant2 1134	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup(𝐵, ℝ^, < ) ∈ ℝ^)
6		elun 4128	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
7		xrltso 13157	. . . . . . . . 9 ⊢ < Or ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
9		xrsupss 13325	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 < 𝑤)))
10	8, 9	supub 9471	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
11	10	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
12		supxrcl 13331	. . . . . . . . . . . . 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 3953	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16	15	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
17		xrlelttr 13172	. . . . . . . . . . . 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 1110	. . . . . 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 13325	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐵 𝑧 < 𝑤)))
27	25, 26	supub 9471	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
28	27	3ad2ant2 1134	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
29	24, 28	jaod 859	. . . 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 3131	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^*, < ) < 𝑥)
32		rexr 11281	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
33		xrsupss 13325	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑥 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑥 → ∃𝑦 ∈ 𝐵 𝑧 < 𝑦)))
34	25, 33	suplub 9472	. . . . . . 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 4158	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
37	36	anim1i 615	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 < 𝑦))
38	37	reximi2 3069	. . . . . 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 3131	. . 3 ⊢ (𝐵 ⊆ ℝ^* → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
42	41	3ad2ant2 1134	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
43		supxr 13329	. 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 838	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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 ∪ cun 3924 ⊆ wss 3926 class class class wbr 5119 Or wor 5560 supcsup 9452 ℝcr 11128 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-sup 9454 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469

This theorem is referenced by: supxrmnf 13333 xpsdsval 24320

W3C validator