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Theorem ovolshftlem2 25413

Description: Lemma for ovolshft 25414. (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 2727	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
9		2fveq3 6896	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑔‘𝑚)) = (1^st ‘(𝑔‘𝑛)))
10	9	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑔‘𝑚)) + 𝐶) = ((1^st ‘(𝑔‘𝑛)) + 𝐶))
11		2fveq3 6896	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑔‘𝑚)) = (2^nd ‘(𝑔‘𝑛)))
12	11	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝑔‘𝑚)) + 𝐶) = ((2^nd ‘(𝑔‘𝑛)) + 𝐶))
13	10, 12	opeq12d 4877	. . . . . . . 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 5255	. . . . . . 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 25377	. . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 217	. . . . . . 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 25412	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ∈ 𝑀)
20		eleq1a 2823	. . . . . 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 3150	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
24	23	ralrimiva 3141	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
25		rabss 4065	. 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 233	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 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 ∩ cin 3943 ⊆ wss 3944 ⟨cop 4630 ∪ cuni 4903 ↦ cmpt 5225 × cxp 5670 ran crn 5673 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1^st c1st 7983 2^nd c2nd 7984 ↑_m cmap 8834 supcsup 9449 ℝcr 11123 1c1 11125 + caddc 11127 ℝ^*cxr 11263 < clt 11264 ≤ cle 11265 − cmin 11460 ℕcn 12228 (,)cioo 13342 seqcseq 13984 abscabs 15199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9451 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-ioo 13346 df-ico 13348 df-fz 13503 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201

This theorem is referenced by: ovolshft 25414

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