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Theorem metider 29265
Description: The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
metider (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)

Proof of Theorem metider
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidss 29262 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
2 xpss 5149 . . . 4 (𝑋 × 𝑋) ⊆ (V × V)
31, 2syl6ss 3580 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
4 df-rel 5045 . . 3 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
53, 4sylibr 223 . 2 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
61ssbrd 4626 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑦𝑥(𝑋 × 𝑋)𝑦))
76imp 444 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → 𝑥(𝑋 × 𝑋)𝑦)
8 brxp 5071 . . . 4 (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥𝑋𝑦𝑋))
97, 8sylib 207 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → (𝑥𝑋𝑦𝑋))
10 psmetsym 21925 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
11103expb 1258 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
1211eqeq1d 2612 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ (𝑦𝐷𝑥) = 0))
13 metidv 29263 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
14 metidv 29263 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1514ancom2s 840 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1612, 13, 153bitr4d 299 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑥))
1716biimpd 218 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑥))
1817impancom 455 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → ((𝑥𝑋𝑦𝑋) → 𝑦(~Met𝐷)𝑥))
199, 18mpd 15 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → 𝑦(~Met𝐷)𝑥)
20 simpl 472 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝐷 ∈ (PsMet‘𝑋))
21 simprr 792 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦(~Met𝐷)𝑧)
221ssbrd 4626 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝑦(~Met𝐷)𝑧𝑦(𝑋 × 𝑋)𝑧))
2322imp 444 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → 𝑦(𝑋 × 𝑋)𝑧)
24 brxp 5071 . . . . . . . . 9 (𝑦(𝑋 × 𝑋)𝑧 ↔ (𝑦𝑋𝑧𝑋))
2523, 24sylib 207 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → (𝑦𝑋𝑧𝑋))
2621, 25syldan 486 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝑋𝑧𝑋))
2726simpld 474 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦𝑋)
28 simprl 790 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑦)
2928, 9syldan 486 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝑋𝑦𝑋))
3029simpld 474 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥𝑋)
3126simprd 478 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑧𝑋)
32 psmettri2 21924 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋𝑧𝑋)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3320, 27, 30, 31, 32syl13anc 1320 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3429, 11syldan 486 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
3529, 13syldan 486 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
3628, 35mpbid 221 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = 0)
3734, 36eqtr3d 2646 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑥) = 0)
38 metidv 29263 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
3926, 38syldan 486 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
4021, 39mpbid 221 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑧) = 0)
4137, 40oveq12d 6567 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = (0 +𝑒 0))
42 0xr 9965 . . . . . . 7 0 ∈ ℝ*
43 xaddid1 11946 . . . . . . 7 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4442, 43ax-mp 5 . . . . . 6 (0 +𝑒 0) = 0
4541, 44syl6eq 2660 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = 0)
4633, 45breqtrd 4609 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ 0)
47 psmetge0 21927 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → 0 ≤ (𝑥𝐷𝑧))
4820, 30, 31, 47syl3anc 1318 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 0 ≤ (𝑥𝐷𝑧))
49 psmetcl 21922 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → (𝑥𝐷𝑧) ∈ ℝ*)
5020, 30, 31, 49syl3anc 1318 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ∈ ℝ*)
51 xrletri3 11861 . . . . 5 (((𝑥𝐷𝑧) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5250, 42, 51sylancl 693 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5346, 48, 52mpbir2and 959 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) = 0)
54 metidv 29263 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑧𝑋)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5520, 30, 31, 54syl12anc 1316 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5653, 55mpbird 246 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑧)
57 psmet0 21923 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐷𝑥) = 0)
58 metidv 29263 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑥𝑋)) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
5958anabsan2 859 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
6057, 59mpbird 246 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥(~Met𝐷)𝑥)
611ssbrd 4626 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑥𝑥(𝑋 × 𝑋)𝑥))
6261imp 444 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥(𝑋 × 𝑋)𝑥)
63 brxp 5071 . . . . 5 (𝑥(𝑋 × 𝑋)𝑥 ↔ (𝑥𝑋𝑥𝑋))
6462, 63sylib 207 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → (𝑥𝑋𝑥𝑋))
6564simpld 474 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥𝑋)
6660, 65impbida 873 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋𝑥(~Met𝐷)𝑥))
675, 19, 56, 66iserd 7655 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540   class class class wbr 4583   × cxp 5036  Rel wrel 5043  cfv 5804  (class class class)co 6549   Er wer 7626  0cc0 9815  *cxr 9952  cle 9954   +𝑒 cxad 11820  PsMetcpsmet 19551  ~Metcmetid 29257
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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-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-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  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-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-psmet 19559  df-metid 29259
This theorem is referenced by:  pstmxmet  29268
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