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Theorem metideq 30451
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 475 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2 metidss 30449 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
3 dmss 5527 . . . . . . . . 9 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ dom (𝑋 × 𝑋))
5 dmxpid 5549 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
64, 5syl6sseq 3848 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → dom (~Met𝐷) ⊆ 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → dom (~Met𝐷) ⊆ 𝑋)
8 xpss 5329 . . . . . . . . . 10 (𝑋 × 𝑋) ⊆ (V × V)
92, 8syl6ss 3811 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (V × V))
10 df-rel 5320 . . . . . . . . 9 (Rel (~Met𝐷) ↔ (~Met𝐷) ⊆ (V × V))
119, 10sylibr 226 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met𝐷))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → Rel (~Met𝐷))
13 simprl 788 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴(~Met𝐷)𝐵)
14 releldm 5563 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐴 ∈ dom (~Met𝐷))
1512, 13, 14syl2anc 580 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴 ∈ dom (~Met𝐷))
167, 15sseldd 3800 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐴𝑋)
17 simprr 790 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸(~Met𝐷)𝐹)
18 releldm 5563 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐸 ∈ dom (~Met𝐷))
1912, 17, 18syl2anc 580 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸 ∈ dom (~Met𝐷))
207, 19sseldd 3800 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐸𝑋)
21 psmetsym 22442 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐸𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1491 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 22438 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2423fovrnda 7040 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋)) → (𝐸𝐷𝐴) ∈ ℝ*)
251, 20, 16, 24syl12anc 866 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐴) ∈ ℝ*)
2622, 25eqeltrd 2879 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5558 . . . . . . . 8 ((~Met𝐷) ⊆ (𝑋 × 𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ ran (𝑋 × 𝑋))
29 rnxpid 5785 . . . . . . 7 ran (𝑋 × 𝑋) = 𝑋
3028, 29syl6sseq 3848 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → ran (~Met𝐷) ⊆ 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ran (~Met𝐷) ⊆ 𝑋)
32 relelrn 5564 . . . . . 6 ((Rel (~Met𝐷) ∧ 𝐴(~Met𝐷)𝐵) → 𝐵 ∈ ran (~Met𝐷))
3312, 13, 32syl2anc 580 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵 ∈ ran (~Met𝐷))
3431, 33sseldd 3800 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐵𝑋)
3523fovrnda 7040 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ∈ ℝ*)
361, 34, 20, 35syl12anc 866 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ∈ ℝ*)
37 relelrn 5564 . . . . . . 7 ((Rel (~Met𝐷) ∧ 𝐸(~Met𝐷)𝐹) → 𝐹 ∈ ran (~Met𝐷))
3812, 17, 37syl2anc 580 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹 ∈ ran (~Met𝐷))
3931, 38sseldd 3800 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → 𝐹𝑋)
40 psmetsym 22442 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐵𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
411, 39, 34, 40syl3anc 1491 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
4223fovrnda 7040 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋)) → (𝐹𝐷𝐵) ∈ ℝ*)
431, 39, 34, 42syl12anc 866 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐵) ∈ ℝ*)
4441, 43eqeltrrd 2880 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45 psmettri2 22441 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐴𝑋𝐸𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1492 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47 psmetsym 22442 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
481, 16, 34, 47syl3anc 1491 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
4916, 34jca 508 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝑋𝐵𝑋))
50 metidv 30450 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
5150biimpa 469 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴(~Met𝐷)𝐵) → (𝐴𝐷𝐵) = 0)
521, 49, 13, 51syl21anc 867 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐵) = 0)
5348, 52eqtr3d 2836 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
5453oveq1d 6894 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55 xaddid2 12321 . . . . . 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 2834 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
5846, 57breqtrd 4870 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59 psmettri2 22441 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹𝑋𝐵𝑋𝐸𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1492 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 22442 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹𝑋𝐸𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1491 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 508 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝑋𝐹𝑋))
64 metidv 30450 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) → (𝐸(~Met𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 469 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐹𝑋)) ∧ 𝐸(~Met𝐷)𝐹) → (𝐸𝐷𝐹) = 0)
661, 63, 17, 65syl21anc 867 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐸𝐷𝐹) = 0)
6762, 66eqtrd 2834 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
6867oveq2d 6895 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69 xaddid1 12320 . . . . . 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 2838 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
7260, 71breqtrd 4870 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 12241 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
7423fovrnda 7040 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ∈ ℝ*)
751, 16, 39, 74syl12anc 866 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ∈ ℝ*)
76 psmettri2 22441 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐹𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1492 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 6894 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddid2 12321 . . . . . 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 2834 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 4870 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83 psmettri2 22441 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸𝑋𝐴𝑋𝐹𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1492 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddid1 12320 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 6895 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2844 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 4870 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 12241 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91 xrletri3 12233 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐵𝐷𝐹) ∈ ℝ*) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 580 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 705 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  wss 3770   class class class wbr 4844   × cxp 5311  dom cdm 5313  ran crn 5314  Rel wrel 5318  cfv 6102  (class class class)co 6879  0cc0 10225  *cxr 10363  cle 10365   +𝑒 cxad 12190  PsMetcpsmet 20051  ~Metcmetid 30444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-po 5234  df-so 5235  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-er 7983  df-map 8098  df-en 8197  df-dom 8198  df-sdom 8199  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-xadd 12193  df-psmet 20059  df-metid 30446
This theorem is referenced by:  pstmfval  30454
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