supxrun - Metamath Proof Explorer

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Theorem supxrun 13295

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 4185	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ^)
2	1	biimpi 215	. . 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 13294	. . 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 4149	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
7		xrltso 13120	. . . . . . . . 9 ⊢ < Or ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
9		xrsupss 13288	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 < 𝑤)))
10	8, 9	supub 9454	. . . . . . 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 13294	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
13	12	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
14	4	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐵, ℝ^, < ) ∈ ℝ^*)
15		ssel2 3978	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16	15	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
17		xrlelttr 13135	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → ((sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ∧ sup(𝐵, ℝ^, < ) < 𝑥) → sup(𝐴, ℝ^*, < ) < 𝑥))
19	18	expdimp 454	. . . . . . . . . 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 436	. . . . . . . 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 1112	. . . . . 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 13288	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐵 𝑧 < 𝑤)))
27	25, 26	supub 9454	. . . . . 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 858	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
30	6, 29	biimtrid 241	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ (𝐴 ∪ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
31	30	ralrimiv 3146	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^*, < ) < 𝑥)
32		rexr 11260	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
33		xrsupss 13288	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑥 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑥 → ∃𝑦 ∈ 𝐵 𝑧 < 𝑦)))
34	25, 33	suplub 9455	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
35	32, 34	sylani 605	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
36		elun2 4178	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
37	36	anim1i 616	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 < 𝑦))
38	37	reximi2 3080	. . . . . 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 417	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ ℝ → (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦)))
41	40	ralrimiv 3146	. . 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 13292	. 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 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∪ cun 3947 ⊆ wss 3949 class class class wbr 5149 Or wor 5588 supcsup 9435 ℝcr 11109 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447

This theorem is referenced by: supxrmnf 13296 xpsdsval 23887

W3C validator