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 13416

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 4135	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ^)
2	1	biimpi 219	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐴 ∪ 𝐵) ⊆ ℝ^)
3	2	3adant3 1150	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝐴 ∪ 𝐵) ⊆ ℝ^*)
4		supxrcl 13415	. . 3 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
5	4	3ad2ant2 1152	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup(𝐵, ℝ^, < ) ∈ ℝ^)
6		elun 4099	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
7		xrltso 13240	. . . . . . . . 9 ⊢ < Or ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
9		xrsupss 13409	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 < 𝑤)))
10	8, 9	supub 9429	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
11	10	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℝ^*, < ) < 𝑥))
12		supxrcl 13415	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
13	12	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
14	4	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → sup(𝐵, ℝ^, < ) ∈ ℝ^*)
15		ssel2 3925	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16	15	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
17		xrlelttr 13255	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → ((sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ∧ sup(𝐵, ℝ^, < ) < 𝑥) → sup(𝐴, ℝ^*, < ) < 𝑥))
19	18	expdimp 458	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (sup(𝐵, ℝ^, < ) < 𝑥 → sup(𝐴, ℝ^*, < ) < 𝑥))
20	19	con3d 153	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
21	20	exp41 440	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (𝐵 ⊆ ℝ^* → (𝑥 ∈ 𝐴 → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^, < ) < 𝑥)))))
22	21	com34 92	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝐵 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) → (𝑥 ∈ 𝐴 → (¬ sup(𝐴, ℝ^, < ) < 𝑥 → ¬ sup(𝐵, ℝ^, < ) < 𝑥)))))
23	22	3imp 1128	. . . . . 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 44	. . . . 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 13409	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑦 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑦 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑤 ∈ 𝐵 𝑧 < 𝑤)))
27	25, 26	supub 9429	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
28	27	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ 𝐵 → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
29	24, 28	jaod 873	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
30	6, 29	biimtrid 245	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → (𝑥 ∈ (𝐴 ∪ 𝐵) → ¬ sup(𝐵, ℝ^*, < ) < 𝑥))
31	30	ralrimiv 3153	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^*, < ) < 𝑥)
32		rexr 11327	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
33		xrsupss 13409	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑧 ∈ 𝐵 ¬ 𝑥 < 𝑧 ∧ ∀𝑧 ∈ ℝ^* (𝑧 < 𝑥 → ∃𝑦 ∈ 𝐵 𝑧 < 𝑦)))
34	25, 33	suplub 9430	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
35	32, 34	sylani 616	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ 𝐵 𝑥 < 𝑦))
36		elun2 4128	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
37	36	anim1i 627	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 < 𝑦))
38	37	reximi2 3095	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑥 < 𝑦 → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦)
39	35, 38	syl6 36	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → ((𝑥 ∈ ℝ ∧ 𝑥 < sup(𝐵, ℝ^*, < )) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
40	39	expd 421	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → (𝑥 ∈ ℝ → (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦)))
41	40	ralrimiv 3153	. . 3 ⊢ (𝐵 ⊆ ℝ^* → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
42	41	3ad2ant2 1152	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ ℝ (𝑥 < sup(𝐵, ℝ^*, < ) → ∃𝑦 ∈ (𝐴 ∪ 𝐵)𝑥 < 𝑦))
43		supxr 13413	. 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 852	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → sup((𝐴 ∪ 𝐵), ℝ^, < ) = sup(𝐵, ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∪ cun 3896 ⊆ wss 3898 class class class wbr 5102 Or wor 5554 supcsup 9410 ℝcr 11171 ℝ^*cxr 11314 < clt 11315 ≤ cle 11316

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249 ax-pre-sup 11250

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-po 5555 df-so 5556 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-sup 9412 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516

This theorem is used by: supxrmnf 13417 xpsdsval 24662

W3C validator