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Theorem metider 30277
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 30274 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
2 xpss 5265 . . . 4 (𝑋 × 𝑋) ⊆ (V × V)
31, 2syl6ss 3764 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
4 df-rel 5256 . . 3 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
53, 4sylibr 224 . 2 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
61ssbrd 4829 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑦𝑥(𝑋 × 𝑋)𝑦))
76imp 393 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → 𝑥(𝑋 × 𝑋)𝑦)
8 brxp 5287 . . . 4 (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥𝑋𝑦𝑋))
97, 8sylib 208 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → (𝑥𝑋𝑦𝑋))
10 psmetsym 22335 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
11103expb 1113 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
1211eqeq1d 2773 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ (𝑦𝐷𝑥) = 0))
13 metidv 30275 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
14 metidv 30275 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1514ancom2s 629 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1612, 13, 153bitr4d 300 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑥))
1716biimpd 219 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑥))
1817impancom 439 . . 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 468 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝐷 ∈ (PsMet‘𝑋))
21 simprr 756 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦(~Met𝐷)𝑧)
221ssbrd 4829 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝑦(~Met𝐷)𝑧𝑦(𝑋 × 𝑋)𝑧))
2322imp 393 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → 𝑦(𝑋 × 𝑋)𝑧)
24 brxp 5287 . . . . . . . . 9 (𝑦(𝑋 × 𝑋)𝑧 ↔ (𝑦𝑋𝑧𝑋))
2523, 24sylib 208 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → (𝑦𝑋𝑧𝑋))
2621, 25syldan 579 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝑋𝑧𝑋))
2726simpld 482 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦𝑋)
28 simprl 754 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑦)
2928, 9syldan 579 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝑋𝑦𝑋))
3029simpld 482 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥𝑋)
3126simprd 483 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑧𝑋)
32 psmettri2 22334 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋𝑧𝑋)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3320, 27, 30, 31, 32syl13anc 1478 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3429, 11syldan 579 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
3529, 13syldan 579 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
3628, 35mpbid 222 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = 0)
3734, 36eqtr3d 2807 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑥) = 0)
38 metidv 30275 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
3926, 38syldan 579 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
4021, 39mpbid 222 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑧) = 0)
4137, 40oveq12d 6811 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = (0 +𝑒 0))
42 0xr 10288 . . . . . . 7 0 ∈ ℝ*
43 xaddid1 12277 . . . . . . 7 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4442, 43ax-mp 5 . . . . . 6 (0 +𝑒 0) = 0
4541, 44syl6eq 2821 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = 0)
4633, 45breqtrd 4812 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ 0)
47 psmetge0 22337 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → 0 ≤ (𝑥𝐷𝑧))
4820, 30, 31, 47syl3anc 1476 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 0 ≤ (𝑥𝐷𝑧))
49 psmetcl 22332 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → (𝑥𝐷𝑧) ∈ ℝ*)
5020, 30, 31, 49syl3anc 1476 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ∈ ℝ*)
51 xrletri3 12190 . . . . 5 (((𝑥𝐷𝑧) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5250, 42, 51sylancl 574 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5346, 48, 52mpbir2and 692 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) = 0)
54 metidv 30275 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑧𝑋)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5520, 30, 31, 54syl12anc 1474 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5653, 55mpbird 247 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑧)
57 psmet0 22333 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐷𝑥) = 0)
58 metidv 30275 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑥𝑋)) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
5958anabsan2 653 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
6057, 59mpbird 247 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥(~Met𝐷)𝑥)
611ssbrd 4829 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑥𝑥(𝑋 × 𝑋)𝑥))
6261imp 393 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥(𝑋 × 𝑋)𝑥)
63 brxp 5287 . . . . 5 (𝑥(𝑋 × 𝑋)𝑥 ↔ (𝑥𝑋𝑥𝑋))
6462, 63sylib 208 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → (𝑥𝑋𝑥𝑋))
6564simpld 482 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥𝑋)
6660, 65impbida 802 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋𝑥(~Met𝐷)𝑥))
675, 19, 56, 66iserd 7922 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723   class class class wbr 4786   × cxp 5247  Rel wrel 5254  cfv 6031  (class class class)co 6793   Er wer 7893  0cc0 10138  *cxr 10275  cle 10277   +𝑒 cxad 12149  PsMetcpsmet 19945  ~Metcmetid 30269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-2 11281  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-psmet 19953  df-metid 30271
This theorem is referenced by:  pstmxmet  30280
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