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Theorem ovolshftlem2 25387

Description: Lemma for ovolshft 25388. (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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ℝ)
3		ovolshft.2	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	ad3antrrr 730	. . . . . . 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 730	. . . . . . 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 2729	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
9		2fveq3 6845	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑔‘𝑚)) = (1^st ‘(𝑔‘𝑛)))
10	9	oveq1d 7384	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑔‘𝑚)) + 𝐶) = ((1^st ‘(𝑔‘𝑛)) + 𝐶))
11		2fveq3 6845	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑔‘𝑚)) = (2^nd ‘(𝑔‘𝑛)))
12	11	oveq1d 7384	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝑔‘𝑚)) + 𝐶) = ((2^nd ‘(𝑔‘𝑛)) + 𝐶))
13	10, 12	opeq12d 4841	. . . . . . . 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 5206	. . . . . . 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 25351	. . . . . . . 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 25386	. . . . . 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 3134	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
24	23	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
25		rabss 4031	. 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 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3402 ∩ cin 3910 ⊆ wss 3911 ⟨cop 4591 ∪ cuni 4867 ↦ cmpt 5183 × cxp 5629 ran crn 5632 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 ↑_m cmap 8776 supcsup 9367 ℝcr 11043 1c1 11045 + caddc 11047 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 ℕcn 12162 (,)cioo 13282 seqcseq 13942 abscabs 15176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-ioo 13286 df-ico 13288 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178

This theorem is referenced by: ovolshft 25388

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