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Theorem vitalilem1OLD 23183
 Description: Obsolete proof of vitalilem1 23182 as of 1-May-2021. Lemma for vitali 23188. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
vitali.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
Assertion
Ref Expression
vitalilem1OLD Er (0[,]1)
Distinct variable group:   𝑥,𝑦,

Proof of Theorem vitalilem1OLD
Dummy variables 𝑣 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vitali.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
21relopabi 5167 . . . 4 Rel
32a1i 11 . . 3 (⊤ → Rel )
4 simplr 788 . . . . . 6 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → 𝑣 ∈ (0[,]1))
5 simpll 786 . . . . . 6 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → 𝑢 ∈ (0[,]1))
6 unitssre 12190 . . . . . . . . . . 11 (0[,]1) ⊆ ℝ
76sseli 3564 . . . . . . . . . 10 (𝑢 ∈ (0[,]1) → 𝑢 ∈ ℝ)
87recnd 9947 . . . . . . . . 9 (𝑢 ∈ (0[,]1) → 𝑢 ∈ ℂ)
98ad2antrr 758 . . . . . . . 8 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → 𝑢 ∈ ℂ)
106sseli 3564 . . . . . . . . . 10 (𝑣 ∈ (0[,]1) → 𝑣 ∈ ℝ)
1110recnd 9947 . . . . . . . . 9 (𝑣 ∈ (0[,]1) → 𝑣 ∈ ℂ)
1211ad2antlr 759 . . . . . . . 8 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → 𝑣 ∈ ℂ)
139, 12negsubdi2d 10287 . . . . . . 7 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → -(𝑢𝑣) = (𝑣𝑢))
14 qnegcl 11681 . . . . . . . 8 ((𝑢𝑣) ∈ ℚ → -(𝑢𝑣) ∈ ℚ)
1514adantl 481 . . . . . . 7 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → -(𝑢𝑣) ∈ ℚ)
1613, 15eqeltrrd 2689 . . . . . 6 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → (𝑣𝑢) ∈ ℚ)
174, 5, 16jca31 555 . . . . 5 (((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ) → ((𝑣 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) ∧ (𝑣𝑢) ∈ ℚ))
18 oveq12 6558 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥𝑦) = (𝑢𝑣))
1918eleq1d 2672 . . . . . 6 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑦) ∈ ℚ ↔ (𝑢𝑣) ∈ ℚ))
2019, 1brab2ga 5117 . . . . 5 (𝑢 𝑣 ↔ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ))
21 oveq12 6558 . . . . . . 7 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝑥𝑦) = (𝑣𝑢))
2221eleq1d 2672 . . . . . 6 ((𝑥 = 𝑣𝑦 = 𝑢) → ((𝑥𝑦) ∈ ℚ ↔ (𝑣𝑢) ∈ ℚ))
2322, 1brab2ga 5117 . . . . 5 (𝑣 𝑢 ↔ ((𝑣 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) ∧ (𝑣𝑢) ∈ ℚ))
2417, 20, 233imtr4i 280 . . . 4 (𝑢 𝑣𝑣 𝑢)
2524adantl 481 . . 3 ((⊤ ∧ 𝑢 𝑣) → 𝑣 𝑢)
26 simpl 472 . . . . . . . . 9 ((𝑢 𝑣𝑣 𝑤) → 𝑢 𝑣)
2726, 20sylib 207 . . . . . . . 8 ((𝑢 𝑣𝑣 𝑤) → ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) ∧ (𝑢𝑣) ∈ ℚ))
2827simpld 474 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → (𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)))
2928simpld 474 . . . . . 6 ((𝑢 𝑣𝑣 𝑤) → 𝑢 ∈ (0[,]1))
30 simpr 476 . . . . . . . . 9 ((𝑢 𝑣𝑣 𝑤) → 𝑣 𝑤)
31 oveq12 6558 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝑥𝑦) = (𝑣𝑤))
3231eleq1d 2672 . . . . . . . . . 10 ((𝑥 = 𝑣𝑦 = 𝑤) → ((𝑥𝑦) ∈ ℚ ↔ (𝑣𝑤) ∈ ℚ))
3332, 1brab2ga 5117 . . . . . . . . 9 (𝑣 𝑤 ↔ ((𝑣 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) ∧ (𝑣𝑤) ∈ ℚ))
3430, 33sylib 207 . . . . . . . 8 ((𝑢 𝑣𝑣 𝑤) → ((𝑣 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) ∧ (𝑣𝑤) ∈ ℚ))
3534simpld 474 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → (𝑣 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)))
3635simprd 478 . . . . . 6 ((𝑢 𝑣𝑣 𝑤) → 𝑤 ∈ (0[,]1))
3729, 36jca 553 . . . . 5 ((𝑢 𝑣𝑣 𝑤) → (𝑢 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)))
3829, 8syl 17 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → 𝑢 ∈ ℂ)
3927, 12syl 17 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → 𝑣 ∈ ℂ)
406, 36sseldi 3566 . . . . . . . 8 ((𝑢 𝑣𝑣 𝑤) → 𝑤 ∈ ℝ)
4140recnd 9947 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → 𝑤 ∈ ℂ)
4238, 39, 41npncand 10295 . . . . . 6 ((𝑢 𝑣𝑣 𝑤) → ((𝑢𝑣) + (𝑣𝑤)) = (𝑢𝑤))
4327simprd 478 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → (𝑢𝑣) ∈ ℚ)
4434simprd 478 . . . . . . 7 ((𝑢 𝑣𝑣 𝑤) → (𝑣𝑤) ∈ ℚ)
45 qaddcl 11680 . . . . . . 7 (((𝑢𝑣) ∈ ℚ ∧ (𝑣𝑤) ∈ ℚ) → ((𝑢𝑣) + (𝑣𝑤)) ∈ ℚ)
4643, 44, 45syl2anc 691 . . . . . 6 ((𝑢 𝑣𝑣 𝑤) → ((𝑢𝑣) + (𝑣𝑤)) ∈ ℚ)
4742, 46eqeltrrd 2689 . . . . 5 ((𝑢 𝑣𝑣 𝑤) → (𝑢𝑤) ∈ ℚ)
48 oveq12 6558 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑤) → (𝑥𝑦) = (𝑢𝑤))
4948eleq1d 2672 . . . . . 6 ((𝑥 = 𝑢𝑦 = 𝑤) → ((𝑥𝑦) ∈ ℚ ↔ (𝑢𝑤) ∈ ℚ))
5049, 1brab2ga 5117 . . . . 5 (𝑢 𝑤 ↔ ((𝑢 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) ∧ (𝑢𝑤) ∈ ℚ))
5137, 47, 50sylanbrc 695 . . . 4 ((𝑢 𝑣𝑣 𝑤) → 𝑢 𝑤)
5251adantl 481 . . 3 ((⊤ ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑢 𝑤)
538subidd 10259 . . . . . . . 8 (𝑢 ∈ (0[,]1) → (𝑢𝑢) = 0)
54 0z 11265 . . . . . . . . 9 0 ∈ ℤ
55 zq 11670 . . . . . . . . 9 (0 ∈ ℤ → 0 ∈ ℚ)
5654, 55ax-mp 5 . . . . . . . 8 0 ∈ ℚ
5753, 56syl6eqel 2696 . . . . . . 7 (𝑢 ∈ (0[,]1) → (𝑢𝑢) ∈ ℚ)
5857adantr 480 . . . . . 6 ((𝑢 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) → (𝑢𝑢) ∈ ℚ)
5958pm4.71i 662 . . . . 5 ((𝑢 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) ↔ ((𝑢 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) ∧ (𝑢𝑢) ∈ ℚ))
60 pm4.24 673 . . . . 5 (𝑢 ∈ (0[,]1) ↔ (𝑢 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)))
61 oveq12 6558 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥𝑦) = (𝑢𝑢))
6261eleq1d 2672 . . . . . 6 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝑥𝑦) ∈ ℚ ↔ (𝑢𝑢) ∈ ℚ))
6362, 1brab2ga 5117 . . . . 5 (𝑢 𝑢 ↔ ((𝑢 ∈ (0[,]1) ∧ 𝑢 ∈ (0[,]1)) ∧ (𝑢𝑢) ∈ ℚ))
6459, 60, 633bitr4i 291 . . . 4 (𝑢 ∈ (0[,]1) ↔ 𝑢 𝑢)
6564a1i 11 . . 3 (⊤ → (𝑢 ∈ (0[,]1) ↔ 𝑢 𝑢))
663, 25, 52, 65iserd 7655 . 2 (⊤ → Er (0[,]1))
6766trud 1484 1 Er (0[,]1)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   class class class wbr 4583  {copab 4642  Rel wrel 5043  (class class class)co 6549   Er wer 7626  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  -cneg 10146  ℤcz 11254  ℚcq 11664  [,]cicc 12049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-q 11665  df-icc 12053 This theorem is referenced by: (None)
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