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Theorem vitalilem2 25358
Description: Lemma for vitali 25362. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1 ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (π‘₯ βˆ’ 𝑦) ∈ β„š)}
vitali.2 𝑆 = ((0[,]1) / ∼ )
vitali.3 (πœ‘ β†’ 𝐹 Fn 𝑆)
vitali.4 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧))
vitali.5 (πœ‘ β†’ 𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)))
vitali.6 𝑇 = (𝑛 ∈ β„• ↦ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹})
vitali.7 (πœ‘ β†’ Β¬ ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol))
Assertion
Ref Expression
vitalilem2 (πœ‘ β†’ (ran 𝐹 βŠ† (0[,]1) ∧ (0[,]1) βŠ† βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∧ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2)))
Distinct variable groups:   π‘š,𝑛,𝑠,π‘₯,𝑦,𝑧,𝐺   πœ‘,π‘š,𝑛,π‘₯,𝑧   𝑧,𝑆   𝑇,π‘š,π‘₯   π‘š,𝐹,𝑛,𝑠,π‘₯,𝑦,𝑧   ∼ ,π‘š,𝑛,𝑠,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(𝑦,𝑠)   𝑆(π‘₯,𝑦,π‘š,𝑛,𝑠)   𝑇(𝑦,𝑧,𝑛,𝑠)

Proof of Theorem vitalilem2
Dummy variables 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vitali.3 . . . 4 (πœ‘ β†’ 𝐹 Fn 𝑆)
2 vitali.4 . . . . 5 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧))
3 vitali.2 . . . . . . . . 9 𝑆 = ((0[,]1) / ∼ )
4 neeq1 3001 . . . . . . . . 9 ([𝑣] ∼ = 𝑧 β†’ ([𝑣] ∼ β‰  βˆ… ↔ 𝑧 β‰  βˆ…))
5 vitali.1 . . . . . . . . . . . . . 14 ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (π‘₯ βˆ’ 𝑦) ∈ β„š)}
65vitalilem1 25357 . . . . . . . . . . . . 13 ∼ Er (0[,]1)
7 erdm 8715 . . . . . . . . . . . . 13 ( ∼ Er (0[,]1) β†’ dom ∼ = (0[,]1))
86, 7ax-mp 5 . . . . . . . . . . . 12 dom ∼ = (0[,]1)
98eleq2i 2823 . . . . . . . . . . 11 (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
10 ecdmn0 8752 . . . . . . . . . . 11 (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ β‰  βˆ…)
119, 10bitr3i 276 . . . . . . . . . 10 (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ β‰  βˆ…)
1211biimpi 215 . . . . . . . . 9 (𝑣 ∈ (0[,]1) β†’ [𝑣] ∼ β‰  βˆ…)
133, 4, 12ectocl 8781 . . . . . . . 8 (𝑧 ∈ 𝑆 β†’ 𝑧 β‰  βˆ…)
1413adantl 480 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 β‰  βˆ…)
15 sseq1 4006 . . . . . . . . . 10 ([𝑀] ∼ = 𝑧 β†’ ([𝑀] ∼ βŠ† (0[,]1) ↔ 𝑧 βŠ† (0[,]1)))
166a1i 11 . . . . . . . . . . 11 (𝑀 ∈ (0[,]1) β†’ ∼ Er (0[,]1))
1716ecss 8751 . . . . . . . . . 10 (𝑀 ∈ (0[,]1) β†’ [𝑀] ∼ βŠ† (0[,]1))
183, 15, 17ectocl 8781 . . . . . . . . 9 (𝑧 ∈ 𝑆 β†’ 𝑧 βŠ† (0[,]1))
1918adantl 480 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 βŠ† (0[,]1))
2019sseld 3980 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((πΉβ€˜π‘§) ∈ 𝑧 β†’ (πΉβ€˜π‘§) ∈ (0[,]1)))
2114, 20embantd 59 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧) β†’ (πΉβ€˜π‘§) ∈ (0[,]1)))
2221ralimdva 3165 . . . . 5 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧) β†’ βˆ€π‘§ ∈ 𝑆 (πΉβ€˜π‘§) ∈ (0[,]1)))
232, 22mpd 15 . . . 4 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (πΉβ€˜π‘§) ∈ (0[,]1))
24 ffnfv 7119 . . . 4 (𝐹:π‘†βŸΆ(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ βˆ€π‘§ ∈ 𝑆 (πΉβ€˜π‘§) ∈ (0[,]1)))
251, 23, 24sylanbrc 581 . . 3 (πœ‘ β†’ 𝐹:π‘†βŸΆ(0[,]1))
2625frnd 6724 . 2 (πœ‘ β†’ ran 𝐹 βŠ† (0[,]1))
27 vitali.5 . . . . . . . 8 (πœ‘ β†’ 𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)))
2827adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)))
29 f1ocnv 6844 . . . . . . 7 (𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)) β†’ ◑𝐺:(β„š ∩ (-1[,]1))–1-1-ontoβ†’β„•)
30 f1of 6832 . . . . . . 7 (◑𝐺:(β„š ∩ (-1[,]1))–1-1-ontoβ†’β„• β†’ ◑𝐺:(β„š ∩ (-1[,]1))βŸΆβ„•)
3128, 29, 303syl 18 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ◑𝐺:(β„š ∩ (-1[,]1))βŸΆβ„•)
32 simpr 483 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ (0[,]1))
3332, 11sylib 217 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ [𝑣] ∼ β‰  βˆ…)
34 neeq1 3001 . . . . . . . . . . . 12 (𝑧 = [𝑣] ∼ β†’ (𝑧 β‰  βˆ… ↔ [𝑣] ∼ β‰  βˆ…))
35 fveq2 6890 . . . . . . . . . . . . 13 (𝑧 = [𝑣] ∼ β†’ (πΉβ€˜π‘§) = (πΉβ€˜[𝑣] ∼ ))
36 id 22 . . . . . . . . . . . . 13 (𝑧 = [𝑣] ∼ β†’ 𝑧 = [𝑣] ∼ )
3735, 36eleq12d 2825 . . . . . . . . . . . 12 (𝑧 = [𝑣] ∼ β†’ ((πΉβ€˜π‘§) ∈ 𝑧 ↔ (πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ ))
3834, 37imbi12d 343 . . . . . . . . . . 11 (𝑧 = [𝑣] ∼ β†’ ((𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧) ↔ ([𝑣] ∼ β‰  βˆ… β†’ (πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ )))
392adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧))
40 ovex 7444 . . . . . . . . . . . . . . 15 (0[,]1) ∈ V
41 erex 8729 . . . . . . . . . . . . . . 15 ( ∼ Er (0[,]1) β†’ ((0[,]1) ∈ V β†’ ∼ ∈ V))
426, 40, 41mp2 9 . . . . . . . . . . . . . 14 ∼ ∈ V
4342ecelqsi 8769 . . . . . . . . . . . . 13 (𝑣 ∈ (0[,]1) β†’ [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
4443adantl 480 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
4544, 3eleqtrrdi 2842 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ [𝑣] ∼ ∈ 𝑆)
4638, 39, 45rspcdva 3612 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ([𝑣] ∼ β‰  βˆ… β†’ (πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ ))
4733, 46mpd 15 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ )
48 fvex 6903 . . . . . . . . . . 11 (πΉβ€˜[𝑣] ∼ ) ∈ V
49 vex 3476 . . . . . . . . . . 11 𝑣 ∈ V
5048, 49elec 8749 . . . . . . . . . 10 ((πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (πΉβ€˜[𝑣] ∼ ))
51 oveq12 7420 . . . . . . . . . . . 12 ((π‘₯ = 𝑣 ∧ 𝑦 = (πΉβ€˜[𝑣] ∼ )) β†’ (π‘₯ βˆ’ 𝑦) = (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))
5251eleq1d 2816 . . . . . . . . . . 11 ((π‘₯ = 𝑣 ∧ 𝑦 = (πΉβ€˜[𝑣] ∼ )) β†’ ((π‘₯ βˆ’ 𝑦) ∈ β„š ↔ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ β„š))
5352, 5brab2a 5768 . . . . . . . . . 10 (𝑣 ∼ (πΉβ€˜[𝑣] ∼ ) ↔ ((𝑣 ∈ (0[,]1) ∧ (πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ β„š))
5450, 53bitri 274 . . . . . . . . 9 ((πΉβ€˜[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ ((𝑣 ∈ (0[,]1) ∧ (πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ β„š))
5547, 54sylib 217 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ((𝑣 ∈ (0[,]1) ∧ (πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ β„š))
5655simprd 494 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ β„š)
57 elicc01 13447 . . . . . . . . . . 11 (𝑣 ∈ (0[,]1) ↔ (𝑣 ∈ ℝ ∧ 0 ≀ 𝑣 ∧ 𝑣 ≀ 1))
5832, 57sylib 217 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 ∈ ℝ ∧ 0 ≀ 𝑣 ∧ 𝑣 ≀ 1))
5958simp1d 1140 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ ℝ)
6055simpld 493 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 ∈ (0[,]1) ∧ (πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1)))
6160simprd 494 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1))
62 elicc01 13447 . . . . . . . . . . 11 ((πΉβ€˜[𝑣] ∼ ) ∈ (0[,]1) ↔ ((πΉβ€˜[𝑣] ∼ ) ∈ ℝ ∧ 0 ≀ (πΉβ€˜[𝑣] ∼ ) ∧ (πΉβ€˜[𝑣] ∼ ) ≀ 1))
6361, 62sylib 217 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ((πΉβ€˜[𝑣] ∼ ) ∈ ℝ ∧ 0 ≀ (πΉβ€˜[𝑣] ∼ ) ∧ (πΉβ€˜[𝑣] ∼ ) ≀ 1))
6463simp1d 1140 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ ℝ)
6559, 64resubcld 11646 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ ℝ)
6664, 59resubcld 11646 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) ∈ ℝ)
67 1red 11219 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 1 ∈ ℝ)
6858simp2d 1141 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 0 ≀ 𝑣)
6964, 59subge02d 11810 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (0 ≀ 𝑣 ↔ ((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) ≀ (πΉβ€˜[𝑣] ∼ )))
7068, 69mpbid 231 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) ≀ (πΉβ€˜[𝑣] ∼ ))
7163simp3d 1142 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ≀ 1)
7266, 64, 67, 70, 71letrd 11375 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ ((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) ≀ 1)
7366, 67lenegd 11797 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) ≀ 1 ↔ -1 ≀ -((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣)))
7472, 73mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ -1 ≀ -((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣))
7564recnd 11246 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ β„‚)
7659recnd 11246 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ β„‚)
7775, 76negsubdi2d 11591 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ -((πΉβ€˜[𝑣] ∼ ) βˆ’ 𝑣) = (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))
7874, 77breqtrd 5173 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ -1 ≀ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))
7963simp2d 1141 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 0 ≀ (πΉβ€˜[𝑣] ∼ ))
8059, 64subge02d 11810 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (0 ≀ (πΉβ€˜[𝑣] ∼ ) ↔ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ≀ 𝑣))
8179, 80mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ≀ 𝑣)
8258simp3d 1142 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ≀ 1)
8365, 59, 67, 81, 82letrd 11375 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ≀ 1)
84 neg1rr 12331 . . . . . . . . 9 -1 ∈ ℝ
85 1re 11218 . . . . . . . . 9 1 ∈ ℝ
8684, 85elicc2i 13394 . . . . . . . 8 ((𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ (-1[,]1) ↔ ((𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ ℝ ∧ -1 ≀ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∧ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ≀ 1))
8765, 78, 83, 86syl3anbrc 1341 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ (-1[,]1))
8856, 87elind 4193 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ (β„š ∩ (-1[,]1)))
8931, 88ffvelcdmd 7086 . . . . 5 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) ∈ β„•)
90 oveq1 7418 . . . . . . . 8 (𝑠 = 𝑣 β†’ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) = (𝑣 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))))
9190eleq1d 2816 . . . . . . 7 (𝑠 = 𝑣 β†’ ((𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹 ↔ (𝑣 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹))
92 f1ocnvfv2 7277 . . . . . . . . . . 11 ((𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)) ∧ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )) ∈ (β„š ∩ (-1[,]1))) β†’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))) = (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))
9327, 88, 92syl2an2r 681 . . . . . . . . . 10 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))) = (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))
9493oveq2d 7427 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) = (𝑣 βˆ’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))
9576, 75nncand 11580 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) = (πΉβ€˜[𝑣] ∼ ))
9694, 95eqtrd 2770 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) = (πΉβ€˜[𝑣] ∼ ))
97 fnfvelrn 7081 . . . . . . . . 9 ((𝐹 Fn 𝑆 ∧ [𝑣] ∼ ∈ 𝑆) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ ran 𝐹)
981, 45, 97syl2an2r 681 . . . . . . . 8 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (πΉβ€˜[𝑣] ∼ ) ∈ ran 𝐹)
9996, 98eqeltrd 2831 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (𝑣 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹)
10091, 59, 99elrabd 3684 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹})
101 fveq2 6890 . . . . . . . . . . 11 (𝑛 = (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) β†’ (πΊβ€˜π‘›) = (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))))
102101oveq2d 7427 . . . . . . . . . 10 (𝑛 = (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) β†’ (𝑠 βˆ’ (πΊβ€˜π‘›)) = (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))))
103102eleq1d 2816 . . . . . . . . 9 (𝑛 = (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) β†’ ((𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹 ↔ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹))
104103rabbidv 3438 . . . . . . . 8 (𝑛 = (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹})
105 vitali.6 . . . . . . . 8 𝑇 = (𝑛 ∈ β„• ↦ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹})
106 reex 11203 . . . . . . . . 9 ℝ ∈ V
107106rabex 5331 . . . . . . . 8 {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹} ∈ V
108104, 105, 107fvmpt 6997 . . . . . . 7 ((β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) ∈ β„• β†’ (π‘‡β€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹})
10989, 108syl 17 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ (π‘‡β€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) ∈ ran 𝐹})
110100, 109eleqtrrd 2834 . . . . 5 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ (π‘‡β€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))))
111 fveq2 6890 . . . . . 6 (π‘š = (β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) β†’ (π‘‡β€˜π‘š) = (π‘‡β€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ )))))
112111eliuni 5002 . . . . 5 (((β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))) ∈ β„• ∧ 𝑣 ∈ (π‘‡β€˜(β—‘πΊβ€˜(𝑣 βˆ’ (πΉβ€˜[𝑣] ∼ ))))) β†’ 𝑣 ∈ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š))
11389, 110, 112syl2anc 582 . . . 4 ((πœ‘ ∧ 𝑣 ∈ (0[,]1)) β†’ 𝑣 ∈ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š))
114113ex 411 . . 3 (πœ‘ β†’ (𝑣 ∈ (0[,]1) β†’ 𝑣 ∈ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
115114ssrdv 3987 . 2 (πœ‘ β†’ (0[,]1) βŠ† βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š))
116 eliun 5000 . . . 4 (π‘₯ ∈ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ↔ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘‡β€˜π‘š))
117 fveq2 6890 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ (πΊβ€˜π‘›) = (πΊβ€˜π‘š))
118117oveq2d 7427 . . . . . . . . . . . . . 14 (𝑛 = π‘š β†’ (𝑠 βˆ’ (πΊβ€˜π‘›)) = (𝑠 βˆ’ (πΊβ€˜π‘š)))
119118eleq1d 2816 . . . . . . . . . . . . 13 (𝑛 = π‘š β†’ ((𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹 ↔ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
120119rabbidv 3438 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
121106rabex 5331 . . . . . . . . . . . 12 {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹} ∈ V
122120, 105, 121fvmpt 6997 . . . . . . . . . . 11 (π‘š ∈ β„• β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
123122adantl 480 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
124123eleq2d 2817 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘₯ ∈ (π‘‡β€˜π‘š) ↔ π‘₯ ∈ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹}))
125124biimpa 475 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ π‘₯ ∈ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
126 oveq1 7418 . . . . . . . . . 10 (𝑠 = π‘₯ β†’ (𝑠 βˆ’ (πΊβ€˜π‘š)) = (π‘₯ βˆ’ (πΊβ€˜π‘š)))
127126eleq1d 2816 . . . . . . . . 9 (𝑠 = π‘₯ β†’ ((𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹 ↔ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
128127elrab 3682 . . . . . . . 8 (π‘₯ ∈ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹} ↔ (π‘₯ ∈ ℝ ∧ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
129125, 128sylib 217 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (π‘₯ ∈ ℝ ∧ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
130129simpld 493 . . . . . 6 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ π‘₯ ∈ ℝ)
13184a1i 11 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ -1 ∈ ℝ)
132 iccssre 13410 . . . . . . . . . 10 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) β†’ (-1[,]1) βŠ† ℝ)
13384, 85, 132mp2an 688 . . . . . . . . 9 (-1[,]1) βŠ† ℝ
134 f1of 6832 . . . . . . . . . . . 12 (𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)) β†’ 𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)))
13527, 134syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)))
136135ffvelcdmda 7085 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (β„š ∩ (-1[,]1)))
137136elin2d 4198 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (-1[,]1))
138133, 137sselid 3979 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ ℝ)
139138adantr 479 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (πΊβ€˜π‘š) ∈ ℝ)
140137adantr 479 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (πΊβ€˜π‘š) ∈ (-1[,]1))
14184, 85elicc2i 13394 . . . . . . . . 9 ((πΊβ€˜π‘š) ∈ (-1[,]1) ↔ ((πΊβ€˜π‘š) ∈ ℝ ∧ -1 ≀ (πΊβ€˜π‘š) ∧ (πΊβ€˜π‘š) ≀ 1))
142140, 141sylib 217 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((πΊβ€˜π‘š) ∈ ℝ ∧ -1 ≀ (πΊβ€˜π‘š) ∧ (πΊβ€˜π‘š) ≀ 1))
143142simp2d 1141 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ -1 ≀ (πΊβ€˜π‘š))
14426ad2antrr 722 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ran 𝐹 βŠ† (0[,]1))
145129simprd 494 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹)
146144, 145sseldd 3982 . . . . . . . . . 10 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ (0[,]1))
147 elicc01 13447 . . . . . . . . . 10 ((π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ (0[,]1) ↔ ((π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ℝ ∧ 0 ≀ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∧ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ≀ 1))
148146, 147sylib 217 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((π‘₯ βˆ’ (πΊβ€˜π‘š)) ∈ ℝ ∧ 0 ≀ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ∧ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ≀ 1))
149148simp2d 1141 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ 0 ≀ (π‘₯ βˆ’ (πΊβ€˜π‘š)))
150130, 139subge0d 11808 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (0 ≀ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ↔ (πΊβ€˜π‘š) ≀ π‘₯))
151149, 150mpbid 231 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (πΊβ€˜π‘š) ≀ π‘₯)
152131, 139, 130, 143, 151letrd 11375 . . . . . 6 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ -1 ≀ π‘₯)
153 peano2re 11391 . . . . . . . 8 ((πΊβ€˜π‘š) ∈ ℝ β†’ ((πΊβ€˜π‘š) + 1) ∈ ℝ)
154139, 153syl 17 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((πΊβ€˜π‘š) + 1) ∈ ℝ)
155 2re 12290 . . . . . . . 8 2 ∈ ℝ
156155a1i 11 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ 2 ∈ ℝ)
157148simp3d 1142 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (π‘₯ βˆ’ (πΊβ€˜π‘š)) ≀ 1)
158 1red 11219 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ 1 ∈ ℝ)
159130, 139, 158lesubadd2d 11817 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((π‘₯ βˆ’ (πΊβ€˜π‘š)) ≀ 1 ↔ π‘₯ ≀ ((πΊβ€˜π‘š) + 1)))
160157, 159mpbid 231 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ π‘₯ ≀ ((πΊβ€˜π‘š) + 1))
161142simp3d 1142 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ (πΊβ€˜π‘š) ≀ 1)
162139, 158, 158, 161leadd1dd 11832 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((πΊβ€˜π‘š) + 1) ≀ (1 + 1))
163 df-2 12279 . . . . . . . 8 2 = (1 + 1)
164162, 163breqtrrdi 5189 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ ((πΊβ€˜π‘š) + 1) ≀ 2)
165130, 154, 156, 160, 164letrd 11375 . . . . . 6 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ π‘₯ ≀ 2)
16684, 155elicc2i 13394 . . . . . 6 (π‘₯ ∈ (-1[,]2) ↔ (π‘₯ ∈ ℝ ∧ -1 ≀ π‘₯ ∧ π‘₯ ≀ 2))
167130, 152, 165, 166syl3anbrc 1341 . . . . 5 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ (π‘‡β€˜π‘š)) β†’ π‘₯ ∈ (-1[,]2))
168167rexlimdva2 3155 . . . 4 (πœ‘ β†’ (βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘‡β€˜π‘š) β†’ π‘₯ ∈ (-1[,]2)))
169116, 168biimtrid 241 . . 3 (πœ‘ β†’ (π‘₯ ∈ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) β†’ π‘₯ ∈ (-1[,]2)))
170169ssrdv 3987 . 2 (πœ‘ β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2))
17126, 115, 1703jca 1126 1 (πœ‘ β†’ (ran 𝐹 βŠ† (0[,]1) ∧ (0[,]1) βŠ† βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∧ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βˆͺ ciun 4996   class class class wbr 5147  {copab 5209   ↦ cmpt 5230  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   Er wer 8702  [cec 8703   / cqs 8704  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   ≀ cle 11253   βˆ’ cmin 11448  -cneg 11449  β„•cn 12216  2c2 12271  β„šcq 12936  [,]cicc 13331  volcvol 25212
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-ec 8707  df-qs 8711  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-q 12937  df-icc 13335
This theorem is referenced by:  vitalilem3  25359  vitalilem4  25360  vitalilem5  25361
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