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Theorem ovolshftlem2 25452

Description: Lemma for ovolshft 25453. (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 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ℝ)
3		ovolshft.2	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	ad3antrrr 728	. . . . . . 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 728	. . . . . . 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 2725	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
9		2fveq3 6895	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑔‘𝑚)) = (1^st ‘(𝑔‘𝑛)))
10	9	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑔‘𝑚)) + 𝐶) = ((1^st ‘(𝑔‘𝑛)) + 𝐶))
11		2fveq3 6895	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑔‘𝑚)) = (2^nd ‘(𝑔‘𝑛)))
12	11	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝑔‘𝑚)) + 𝐶) = ((2^nd ‘(𝑔‘𝑛)) + 𝐶))
13	10, 12	opeq12d 4878	. . . . . . . 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 5257	. . . . . . 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 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
16		elovolmlem 25416	. . . . . . . 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 483	. . . . . . 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 25451	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ∈ 𝑀)
20		eleq1a 2820	. . . . . 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 452	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
23	22	rexlimdva 3145	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
24	23	ralrimiva 3136	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
25		rabss 4062	. 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ∩ cin 3940 ⊆ wss 3941 ⟨cop 4631 ∪ cuni 4904 ↦ cmpt 5227 × cxp 5671 ran crn 5674 ∘ ccom 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 1^st c1st 7985 2^nd c2nd 7986 ↑_m cmap 8838 supcsup 9458 ℝcr 11132 1c1 11134 + caddc 11136 ℝ^*cxr 11272 < clt 11273 ≤ cle 11274 − cmin 11469 ℕcn 12237 (,)cioo 13351 seqcseq 13993 abscabs 15208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ioo 13355 df-ico 13357 df-fz 13512 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210

This theorem is referenced by: ovolshft 25453

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