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Theorem metideq 32861
Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metideq ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐡𝐷𝐹))
Proof of Theorem metideq
StepHypRef Expression
1 simpl 483 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
2 metidss 32859 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
3 dmss 5900 . . . . . . . . 9 ((~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋) β†’ dom (~Metβ€˜π·) βŠ† dom (𝑋 Γ— 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom (~Metβ€˜π·) βŠ† dom (𝑋 Γ— 𝑋))
5 dmxpid 5927 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
64, 5sseqtrdi 4031 . . . . . . 7 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom (~Metβ€˜π·) βŠ† 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ dom (~Metβ€˜π·) βŠ† 𝑋)
8 xpss 5691 . . . . . . . . . 10 (𝑋 Γ— 𝑋) βŠ† (V Γ— V)
92, 8sstrdi 3993 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (V Γ— V))
10 df-rel 5682 . . . . . . . . 9 (Rel (~Metβ€˜π·) ↔ (~Metβ€˜π·) βŠ† (V Γ— V))
119, 10sylibr 233 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ Rel (~Metβ€˜π·))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ Rel (~Metβ€˜π·))
13 simprl 769 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴(~Metβ€˜π·)𝐡)
14 releldm 5941 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
1512, 13, 14syl2anc 584 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
167, 15sseldd 3982 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴 ∈ 𝑋)
17 simprr 771 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸(~Metβ€˜π·)𝐹)
18 releldm 5941 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
1912, 17, 18syl2anc 584 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
207, 19sseldd 3982 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ 𝑋)
21 psmetsym 23807 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1371 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 23803 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„*)
2423fovcdmda 7574 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (𝐸𝐷𝐴) ∈ ℝ*)
251, 20, 16, 24syl12anc 835 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸𝐷𝐴) ∈ ℝ*)
2622, 25eqeltrd 2833 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5936 . . . . . . . 8 ((~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋) β†’ ran (~Metβ€˜π·) βŠ† ran (𝑋 Γ— 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ran (~Metβ€˜π·) βŠ† ran (𝑋 Γ— 𝑋))
29 rnxpid 6169 . . . . . . 7 ran (𝑋 Γ— 𝑋) = 𝑋
3028, 29sseqtrdi 4031 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ran (~Metβ€˜π·) βŠ† 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ran (~Metβ€˜π·) βŠ† 𝑋)
32 relelrn 5942 . . . . . 6 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3312, 13, 32syl2anc 584 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3431, 33sseldd 3982 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ 𝑋)
3523fovcdmda 7574 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐡𝐷𝐸) ∈ ℝ*)
361, 34, 20, 35syl12anc 835 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ∈ ℝ*)
37 relelrn 5942 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3812, 17, 37syl2anc 584 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3931, 38sseldd 3982 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ 𝑋)
40 psmetsym 23807 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
411, 39, 34, 40syl3anc 1371 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
4223fovcdmda 7574 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐹𝐷𝐡) ∈ ℝ*)
431, 39, 34, 42syl12anc 835 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐡) ∈ ℝ*)
4441, 43eqeltrrd 2834 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ∈ ℝ*)
45 psmettri2 23806 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1372 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
47 psmetsym 23807 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
481, 16, 34, 47syl3anc 1371 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
4916, 34jca 512 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
50 metidv 32860 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴(~Metβ€˜π·)𝐡 ↔ (𝐴𝐷𝐡) = 0))
5150biimpa 477 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ (𝐴𝐷𝐡) = 0)
521, 49, 13, 51syl21anc 836 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐡) = 0)
5348, 52eqtr3d 2774 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐴) = 0)
5453oveq1d 7420 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (0 +𝑒 (𝐡𝐷𝐸)))
55 xaddlid 13217 . . . . . 6 ((𝐡𝐷𝐸) ∈ ℝ* β†’ (0 +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5636, 55syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (0 +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5754, 56eqtrd 2772 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5846, 57breqtrd 5173 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐸))
59 psmettri2 23806 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1372 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 23807 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1371 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 512 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
64 metidv 32860 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐸(~Metβ€˜π·)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 477 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ (𝐸𝐷𝐹) = 0)
661, 63, 17, 65syl21anc 836 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸𝐷𝐹) = 0)
6762, 66eqtrd 2772 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = 0)
6867oveq2d 7421 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐡) +𝑒 0))
69 xaddrid 13216 . . . . . 6 ((𝐹𝐷𝐡) ∈ ℝ* β†’ ((𝐹𝐷𝐡) +𝑒 0) = (𝐹𝐷𝐡))
7043, 69syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 0) = (𝐹𝐷𝐡))
7168, 70, 413eqtrd 2776 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = (𝐡𝐷𝐹))
7260, 71breqtrd 5173 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ (𝐡𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 13137 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹))
7423fovcdmda 7574 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐴𝐷𝐹) ∈ ℝ*)
751, 16, 39, 74syl12anc 835 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ∈ ℝ*)
76 psmettri2 23806 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1372 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 7420 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddlid 13217 . . . . . 6 ((𝐴𝐷𝐹) ∈ ℝ* β†’ (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8075, 79syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8178, 80eqtrd 2772 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 5173 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐹))
83 psmettri2 23806 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1372 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddrid 13216 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* β†’ ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 7421 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2782 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 5173 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 13137 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))
91 xrletri3 13129 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐡𝐷𝐹) ∈ ℝ*) β†’ ((𝐴𝐷𝐸) = (𝐡𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹) ∧ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 584 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐸) = (𝐡𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹) ∧ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 711 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐡𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3947   class class class wbr 5147   Γ— cxp 5673  dom cdm 5675  ran crn 5676  Rel wrel 5680  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  β„*cxr 11243   ≀ cle 11245   +𝑒 cxad 13086  PsMetcpsmet 20920  ~Metcmetid 32854
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-xadd 13089  df-psmet 20928  df-metid 32856
This theorem is referenced by:  pstmfval  32864
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