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Theorem metideq 30245
 Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metideq ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))

Proof of Theorem metideq
StepHypRef Expression
1 simpl 474 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2 metidss 30243 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
3 dmss 5478 . . . . . . . . 9 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
5 dmxpid 5500 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
64, 5syl6sseq 3792 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → dom (~Met𝐷) ⊆ 𝑋)
8 xpss 5282 . . . . . . . . . 10 (𝑋 × 𝑋) ⊆ (V × V)
92, 8syl6ss 3756 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
10 df-rel 5273 . . . . . . . . 9 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
119, 10sylibr 224 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → Rel (~Met𝐷))
13 simprl 811 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴(~Met𝐷)𝐵)
14 releldm 5513 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐴 ∈ dom (~Met𝐷))
1512, 13, 14syl2anc 696 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴 ∈ dom (~Met𝐷))
167, 15sseldd 3745 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴𝑋)
17 simprr 813 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸(~Met𝐷)𝐹)
18 releldm 5513 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐸 ∈ dom (~Met𝐷))
1912, 17, 18syl2anc 696 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸 ∈ dom (~Met𝐷))
207, 19sseldd 3745 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸𝑋)
21 psmetsym 22316 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐸𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 22312 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2423fovrnda 6970 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋)) → (𝐸𝐷𝐴) ∈ ℝ*)
251, 20, 16, 24syl12anc 1475 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐴) ∈ ℝ*)
2622, 25eqeltrd 2839 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5509 . . . . . . . 8 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
29 rnxpid 5725 . . . . . . 7 ran (𝑋 × 𝑋) = 𝑋
3028, 29syl6sseq 3792 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ran (~Met𝐷) ⊆ 𝑋)
32 relelrn 5514 . . . . . 6 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐵 ∈ ran (~Met𝐷))
3312, 13, 32syl2anc 696 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵 ∈ ran (~Met𝐷))
3431, 33sseldd 3745 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵𝑋)
3523fovrnda 6970 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ∈ ℝ*)
361, 34, 20, 35syl12anc 1475 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ∈ ℝ*)
37 relelrn 5514 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐹 ∈ ran (~Met𝐷))
3812, 17, 37syl2anc 696 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹 ∈ ran (~Met𝐷))
3931, 38sseldd 3745 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹𝑋)
40 psmetsym 22316 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐵𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
411, 39, 34, 40syl3anc 1477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
4223fovrnda 6970 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋)) → (𝐹𝐷𝐵) ∈ ℝ*)
431, 39, 34, 42syl12anc 1475 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) ∈ ℝ*)
4441, 43eqeltrrd 2840 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45 psmettri2 22315 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐴𝑋𝐸𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1479 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47 psmetsym 22316 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
481, 16, 34, 47syl3anc 1477 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
4916, 34jca 555 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝑋𝐵𝑋))
50 metidv 30244 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
5150biimpa 502 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴(~Met𝐷)𝐵) → (𝐴𝐷𝐵) = 0)
521, 49, 13, 51syl21anc 1476 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = 0)
5348, 52eqtr3d 2796 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
5453oveq1d 6828 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55 xaddid2 12266 . . . . . 6 ((𝐵𝐷𝐸) ∈ ℝ* → (0 +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5636, 55syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (0 +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5754, 56eqtrd 2794 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5846, 57breqtrd 4830 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59 psmettri2 22315 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1479 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 22316 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐸𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1477 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 555 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝑋𝐹𝑋))
64 metidv 30244 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) → (𝐸(~Met𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 502 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) ∧ 𝐸(~Met𝐷)𝐹) → (𝐸𝐷𝐹) = 0)
661, 63, 17, 65syl21anc 1476 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐹) = 0)
6762, 66eqtrd 2794 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
6867oveq2d 6829 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69 xaddid1 12265 . . . . . 6 ((𝐹𝐷𝐵) ∈ ℝ* → ((𝐹𝐷𝐵) +𝑒 0) = (𝐹𝐷𝐵))
7043, 69syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 0) = (𝐹𝐷𝐵))
7168, 70, 413eqtrd 2798 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
7260, 71breqtrd 4830 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 12186 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
7423fovrnda 6970 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ∈ ℝ*)
751, 16, 39, 74syl12anc 1475 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ∈ ℝ*)
76 psmettri2 22315 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐹𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1479 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 6828 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddid2 12266 . . . . . 6 ((𝐴𝐷𝐹) ∈ ℝ* → (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8075, 79syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8178, 80eqtrd 2794 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 4830 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83 psmettri2 22315 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1479 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddid1 12265 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 6829 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2804 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 4830 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 12186 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91 xrletri3 12178 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐵𝐷𝐹) ∈ ℝ*) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 696 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 995 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715   class class class wbr 4804   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271  ‘cfv 6049  (class class class)co 6813  0cc0 10128  ℝ*cxr 10265   ≤ cle 10267   +𝑒 cxad 12137  PsMetcpsmet 19932  ~Metcmetid 30238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-xadd 12140  df-psmet 19940  df-metid 30240 This theorem is referenced by:  pstmfval  30248
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