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Theorem metideq 33403
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 482 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2 metidss 33401 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
3 dmss 5896 . . . . . . . . 9 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
5 dmxpid 5923 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
64, 5sseqtrdi 4027 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → dom (~Met𝐷) ⊆ 𝑋)
8 xpss 5685 . . . . . . . . . 10 (𝑋 × 𝑋) ⊆ (V × V)
92, 8sstrdi 3989 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
10 df-rel 5676 . . . . . . . . 9 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
119, 10sylibr 233 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → Rel (~Met𝐷))
13 simprl 768 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴(~Met𝐷)𝐵)
14 releldm 5937 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐴 ∈ dom (~Met𝐷))
1512, 13, 14syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴 ∈ dom (~Met𝐷))
167, 15sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴𝑋)
17 simprr 770 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸(~Met𝐷)𝐹)
18 releldm 5937 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐸 ∈ dom (~Met𝐷))
1912, 17, 18syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸 ∈ dom (~Met𝐷))
207, 19sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸𝑋)
21 psmetsym 24171 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐸𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 24167 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2423fovcdmda 7575 . . . . 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 2827 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5932 . . . . . . . 8 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
29 rnxpid 6166 . . . . . . 7 ran (𝑋 × 𝑋) = 𝑋
3028, 29sseqtrdi 4027 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ran (~Met𝐷) ⊆ 𝑋)
32 relelrn 5938 . . . . . 6 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐵 ∈ ran (~Met𝐷))
3312, 13, 32syl2anc 583 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵 ∈ ran (~Met𝐷))
3431, 33sseldd 3978 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵𝑋)
3523fovcdmda 7575 . . . 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 5938 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐹 ∈ ran (~Met𝐷))
3812, 17, 37syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹 ∈ ran (~Met𝐷))
3931, 38sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹𝑋)
40 psmetsym 24171 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐵𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
411, 39, 34, 40syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
4223fovcdmda 7575 . . . . 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 2828 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45 psmettri2 24170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐴𝑋𝐸𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47 psmetsym 24171 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
481, 16, 34, 47syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
4916, 34jca 511 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝑋𝐵𝑋))
50 metidv 33402 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
5150biimpa 476 . . . . . . . 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 2768 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
5453oveq1d 7420 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55 xaddlid 13227 . . . . . 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 2766 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5846, 57breqtrd 5167 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59 psmettri2 24170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 24171 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐸𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 511 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝑋𝐹𝑋))
64 metidv 33402 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) → (𝐸(~Met𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 476 . . . . . . . 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 2766 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
6867oveq2d 7421 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69 xaddrid 13226 . . . . . 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 2770 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
7260, 71breqtrd 5167 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 13147 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
7423fovcdmda 7575 . . . 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 24170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐹𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 7420 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddlid 13227 . . . . . 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 2766 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 5167 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83 psmettri2 24170 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddrid 13226 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 7421 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2776 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 5167 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 13147 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91 xrletri3 13139 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐵𝐷𝐹) ∈ ℝ*) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 583 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 710 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943   class class class wbr 5141   × cxp 5667  dom cdm 5669  ran crn 5670  Rel wrel 5674  cfv 6537  (class class class)co 7405  0cc0 11112  *cxr 11251  cle 11253   +𝑒 cxad 13096  PsMetcpsmet 21224  ~Metcmetid 33396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-xadd 13099  df-psmet 21232  df-metid 33398
This theorem is referenced by:  pstmfval  33406
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