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Theorem metideq 31133
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 485 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2 metidss 31131 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
3 dmss 5770 . . . . . . . . 9 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
5 dmxpid 5799 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
64, 5sseqtrdi 4016 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → dom (~Met𝐷) ⊆ 𝑋)
8 xpss 5570 . . . . . . . . . 10 (𝑋 × 𝑋) ⊆ (V × V)
92, 8sstrdi 3978 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
10 df-rel 5561 . . . . . . . . 9 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
119, 10sylibr 236 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → Rel (~Met𝐷))
13 simprl 769 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴(~Met𝐷)𝐵)
14 releldm 5813 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐴 ∈ dom (~Met𝐷))
1512, 13, 14syl2anc 586 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴 ∈ dom (~Met𝐷))
167, 15sseldd 3967 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴𝑋)
17 simprr 771 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸(~Met𝐷)𝐹)
18 releldm 5813 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐸 ∈ dom (~Met𝐷))
1912, 17, 18syl2anc 586 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸 ∈ dom (~Met𝐷))
207, 19sseldd 3967 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸𝑋)
21 psmetsym 22919 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐸𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1367 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 22915 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2423fovrnda 7318 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋)) → (𝐸𝐷𝐴) ∈ ℝ*)
251, 20, 16, 24syl12anc 834 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐴) ∈ ℝ*)
2622, 25eqeltrd 2913 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5808 . . . . . . . 8 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
29 rnxpid 6029 . . . . . . 7 ran (𝑋 × 𝑋) = 𝑋
3028, 29sseqtrdi 4016 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ran (~Met𝐷) ⊆ 𝑋)
32 relelrn 5814 . . . . . 6 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐵 ∈ ran (~Met𝐷))
3312, 13, 32syl2anc 586 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵 ∈ ran (~Met𝐷))
3431, 33sseldd 3967 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵𝑋)
3523fovrnda 7318 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ∈ ℝ*)
361, 34, 20, 35syl12anc 834 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ∈ ℝ*)
37 relelrn 5814 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐹 ∈ ran (~Met𝐷))
3812, 17, 37syl2anc 586 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹 ∈ ran (~Met𝐷))
3931, 38sseldd 3967 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹𝑋)
40 psmetsym 22919 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐵𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
411, 39, 34, 40syl3anc 1367 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
4223fovrnda 7318 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋)) → (𝐹𝐷𝐵) ∈ ℝ*)
431, 39, 34, 42syl12anc 834 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) ∈ ℝ*)
4441, 43eqeltrrd 2914 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45 psmettri2 22918 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐴𝑋𝐸𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47 psmetsym 22919 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
481, 16, 34, 47syl3anc 1367 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
4916, 34jca 514 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝑋𝐵𝑋))
50 metidv 31132 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
5150biimpa 479 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴(~Met𝐷)𝐵) → (𝐴𝐷𝐵) = 0)
521, 49, 13, 51syl21anc 835 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = 0)
5348, 52eqtr3d 2858 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
5453oveq1d 7170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55 xaddid2 12634 . . . . . 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 2856 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5846, 57breqtrd 5091 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59 psmettri2 22918 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 22919 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐸𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1367 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 514 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝑋𝐹𝑋))
64 metidv 31132 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) → (𝐸(~Met𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 479 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) ∧ 𝐸(~Met𝐷)𝐹) → (𝐸𝐷𝐹) = 0)
661, 63, 17, 65syl21anc 835 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐹) = 0)
6762, 66eqtrd 2856 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
6867oveq2d 7171 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69 xaddid1 12633 . . . . . 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 2860 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
7260, 71breqtrd 5091 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 12554 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
7423fovrnda 7318 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ∈ ℝ*)
751, 16, 39, 74syl12anc 834 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ∈ ℝ*)
76 psmettri2 22918 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐹𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 7170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddid2 12634 . . . . . 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 2856 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 5091 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83 psmettri2 22918 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddid1 12633 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 7171 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2866 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 5091 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 12554 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91 xrletri3 12546 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐵𝐷𝐹) ∈ ℝ*) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 586 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 711 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  wss 3935   class class class wbr 5065   × cxp 5552  dom cdm 5554  ran crn 5555  Rel wrel 5559  cfv 6354  (class class class)co 7155  0cc0 10536  *cxr 10673  cle 10675   +𝑒 cxad 12504  PsMetcpsmet 20528  ~Metcmetid 31126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-xadd 12507  df-psmet 20536  df-metid 31128
This theorem is referenced by:  pstmfval  31136
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