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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 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 2728	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
9		2fveq3 6902	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑔‘𝑚)) = (1^st ‘(𝑔‘𝑛)))
10	9	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑔‘𝑚)) + 𝐶) = ((1^st ‘(𝑔‘𝑛)) + 𝐶))
11		2fveq3 6902	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑔‘𝑚)) = (2^nd ‘(𝑔‘𝑛)))
12	11	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝑔‘𝑚)) + 𝐶) = ((2^nd ‘(𝑔‘𝑛)) + 𝐶))
13	10, 12	opeq12d 4882	. . . . . . . 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 5261	. . . . . . 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 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 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 25451	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ∈ 𝑀)
20		eleq1a 2824	. . . . . 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 3152	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) → 𝑧 ∈ 𝑀))
24	23	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < )) → 𝑧 ∈ 𝑀))
25		rabss 4067	. 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 3058 ∃wrex 3067 {crab 3429 ∩ cin 3946 ⊆ wss 3947 ⟨cop 4635 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5676 ran crn 5679 ∘ ccom 5682 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 1^st c1st 7991 2^nd c2nd 7992 ↑_m cmap 8845 supcsup 9464 ℝcr 11138 1c1 11140 + caddc 11142 ℝ^*cxr 11278 < clt 11279 ≤ cle 11280 − cmin 11475 ℕcn 12243 (,)cioo 13357 seqcseq 13999 abscabs 15214

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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-ioo 13361 df-ico 13363 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216

This theorem is referenced by: ovolshft 25453

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