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Theorem metider 31245
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 31242 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
2 xpss 5539 . . . 4 (𝑋 × 𝑋) ⊆ (V × V)
31, 2sstrdi 3930 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
4 df-rel 5530 . . 3 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
53, 4sylibr 237 . 2 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
61ssbrd 5076 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑦𝑥(𝑋 × 𝑋)𝑦))
76imp 410 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → 𝑥(𝑋 × 𝑋)𝑦)
8 brxp 5569 . . . 4 (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥𝑋𝑦𝑋))
97, 8sylib 221 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑦) → (𝑥𝑋𝑦𝑋))
10 psmetsym 22920 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
11103expb 1117 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
1211eqeq1d 2803 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ (𝑦𝐷𝑥) = 0))
13 metidv 31243 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
14 metidv 31243 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1514ancom2s 649 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑦(~Met𝐷)𝑥 ↔ (𝑦𝐷𝑥) = 0))
1612, 13, 153bitr4d 314 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑥))
1716biimpd 232 . . . 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 486 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝐷 ∈ (PsMet‘𝑋))
21 simprr 772 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦(~Met𝐷)𝑧)
221ssbrd 5076 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝑦(~Met𝐷)𝑧𝑦(𝑋 × 𝑋)𝑧))
2322imp 410 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → 𝑦(𝑋 × 𝑋)𝑧)
24 brxp 5569 . . . . . . . . 9 (𝑦(𝑋 × 𝑋)𝑧 ↔ (𝑦𝑋𝑧𝑋))
2523, 24sylib 221 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦(~Met𝐷)𝑧) → (𝑦𝑋𝑧𝑋))
2621, 25syldan 594 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝑋𝑧𝑋))
2726simpld 498 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑦𝑋)
28 simprl 770 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑦)
2928, 9syldan 594 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝑋𝑦𝑋))
3029simpld 498 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥𝑋)
3126simprd 499 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑧𝑋)
32 psmettri2 22919 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑥𝑋𝑧𝑋)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3320, 27, 30, 31, 32syl13anc 1369 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)))
3429, 11syldan 594 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
3529, 13syldan 594 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑦 ↔ (𝑥𝐷𝑦) = 0))
3628, 35mpbid 235 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑦) = 0)
3734, 36eqtr3d 2838 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑥) = 0)
38 metidv 31243 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
3926, 38syldan 594 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦(~Met𝐷)𝑧 ↔ (𝑦𝐷𝑧) = 0))
4021, 39mpbid 235 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑦𝐷𝑧) = 0)
4137, 40oveq12d 7157 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = (0 +𝑒 0))
42 0xr 10681 . . . . . . 7 0 ∈ ℝ*
43 xaddid1 12626 . . . . . . 7 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4442, 43ax-mp 5 . . . . . 6 (0 +𝑒 0) = 0
4541, 44eqtrdi 2852 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑦𝐷𝑥) +𝑒 (𝑦𝐷𝑧)) = 0)
4633, 45breqtrd 5059 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ≤ 0)
47 psmetge0 22922 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → 0 ≤ (𝑥𝐷𝑧))
4820, 30, 31, 47syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 0 ≤ (𝑥𝐷𝑧))
49 psmetcl 22917 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑧𝑋) → (𝑥𝐷𝑧) ∈ ℝ*)
5020, 30, 31, 49syl3anc 1368 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) ∈ ℝ*)
51 xrletri3 12539 . . . . 5 (((𝑥𝐷𝑧) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5250, 42, 51sylancl 589 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → ((𝑥𝐷𝑧) = 0 ↔ ((𝑥𝐷𝑧) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑧))))
5346, 48, 52mpbir2and 712 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥𝐷𝑧) = 0)
54 metidv 31243 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑧𝑋)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5520, 30, 31, 54syl12anc 835 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → (𝑥(~Met𝐷)𝑧 ↔ (𝑥𝐷𝑧) = 0))
5653, 55mpbird 260 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥(~Met𝐷)𝑦𝑦(~Met𝐷)𝑧)) → 𝑥(~Met𝐷)𝑧)
57 psmet0 22918 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐷𝑥) = 0)
58 metidv 31243 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑥𝑋)) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
5958anabsan2 673 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥(~Met𝐷)𝑥 ↔ (𝑥𝐷𝑥) = 0))
6057, 59mpbird 260 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥(~Met𝐷)𝑥)
611ssbrd 5076 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝑥(~Met𝐷)𝑥𝑥(𝑋 × 𝑋)𝑥))
6261imp 410 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥(𝑋 × 𝑋)𝑥)
63 brxp 5569 . . . . 5 (𝑥(𝑋 × 𝑋)𝑥 ↔ (𝑥𝑋𝑥𝑋))
6462, 63sylib 221 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → (𝑥𝑋𝑥𝑋))
6564simpld 498 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥(~Met𝐷)𝑥) → 𝑥𝑋)
6660, 65impbida 800 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋𝑥(~Met𝐷)𝑥))
675, 19, 56, 66iserd 8302 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  wss 3884   class class class wbr 5033   × cxp 5521  Rel wrel 5528  cfv 6328  (class class class)co 7139   Er wer 8273  0cc0 10530  *cxr 10667  cle 10669   +𝑒 cxad 12497  PsMetcpsmet 20078  ~Metcmetid 31237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-2 11692  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-psmet 20086  df-metid 31239
This theorem is referenced by:  pstmxmet  31248
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