supxrun - Metamath Proof Explorer

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Theorem supxrun 13360

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 4146	. . . 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 13359	. . 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 4110	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
7		xrltso 13184	. . . . . . . . 9 ⊢ < Or ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
9		xrsupss 13353	. . . . . . . 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 13359	. . . . . . . . . . . . 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 3935	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16	15	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
17		xrlelttr 13199	. . . . . . . . . . . 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 13353	. . . . . . 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 3159	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^* ∧ sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < )) → ∀𝑥 ∈ (𝐴 ∪ 𝐵) ¬ sup(𝐵, ℝ^*, < ) < 𝑥)
32		rexr 11273	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
33		xrsupss 13353	. . . . . . . 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 4139	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
37	36	anim1i 627	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 < 𝑦))
38	37	reximi2 3101	. . . . . 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 3159	. . 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 13357	. 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 2146 ∀wral 3082 ∃wrex 3092 ∪ cun 3906 ⊆ wss 3908 class class class wbr 5114 Or wor 5573 supcsup 9410 ℝcr 11117 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-po 5574 df-so 5575 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462

This theorem is used by: supxrmnf 13361 xpsdsval 24568

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