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Theorem ovolshftlem2 25564

Description: Lemma for ovolshft 25565. (Contributed by Mario Carneiro, 22-Mar-2014.)

**Hypotheses**
Ref	Expression
ovolshft.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolshft.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
ovolshft.3	⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
ovolshft.4	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}

**Assertion**
Ref	Expression
ovolshftlem2	⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))} ⊆ 𝑀)

Distinct variable groups: 𝑓,𝑔,𝑥,𝑦,𝑧,𝐴 𝐶,𝑓,𝑔,𝑥,𝑦,𝑧 𝐵,𝑓,𝑔,𝑦,𝑧 𝑔,𝑀,𝑧 𝜑,𝑓,𝑔,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥) 𝑀(𝑥,𝑦,𝑓)

Proof of Theorem ovolshftlem2 Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolshft.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2	1	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ℝ)
3		ovolshft.2	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐶 ∈ ℝ)
5		ovolshft.3	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
6	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
7		ovolshft.4	. . . . . . 7 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
8		eqid 2740	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
9		2fveq3 6925	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑔‘𝑚)) = (1^st ‘(𝑔‘𝑛)))
10	9	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑔‘𝑚)) + 𝐶) = ((1^st ‘(𝑔‘𝑛)) + 𝐶))
11		2fveq3 6925	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑔‘𝑚)) = (2^nd ‘(𝑔‘𝑛)))
12	11	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝑔‘𝑚)) + 𝐶) = ((2^nd ‘(𝑔‘𝑛)) + 𝐶))
13	10, 12	opeq12d 4905	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ⟨((1^st ‘(𝑔‘𝑚)) + 𝐶), ((2^nd ‘(𝑔‘𝑚)) + 𝐶)⟩ = ⟨((1^st ‘(𝑔‘𝑛)) + 𝐶), ((2^nd ‘(𝑔‘𝑛)) + 𝐶)⟩)
14	13	cbvmptv 5279	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝑔‘𝑚)) + 𝐶), ((2^nd ‘(𝑔‘𝑚)) + 𝐶)⟩) = (𝑛 ∈ ℕ ↦ ⟨((1^st ‘(𝑔‘𝑛)) + 𝐶), ((2^nd ‘(𝑔‘𝑛)) + 𝐶)⟩)
15		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
16		elovolmlem 25528	. . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 218	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔))
19	2, 4, 6, 7, 8, 14, 17, 18	ovolshftlem1 25563	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ∈ 𝑀)
20		eleq1a 2839	. . . . . 6 ⊢ (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ∈ 𝑀 → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) → 𝑧 ∈ 𝑀))
21	19, 20	syl 17	. . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) → 𝑧 ∈ 𝑀))
22	21	expimpd 453	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
23	22	rexlimdva 3161	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
24	23	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
25		rabss 4095	. 2 ⊢ ({𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ 𝑀 ↔ ∀𝑧 ∈ ℝ^ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
26	24, 25	sylibr 234	1 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))} ⊆ 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 {crab 3443 ∩ cin 3975 ⊆ wss 3976 ⟨cop 4654 ∪ cuni 4931 ↦ cmpt 5249 × cxp 5698 ran crn 5701 ∘ ccom 5704 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 supcsup 9509 ℝcr 11183 1c1 11185 + caddc 11187 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 (,)cioo 13407 seqcseq 14052 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ioo 13411 df-ico 13413 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: ovolshft 25565

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