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Theorem metideq 33527
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 481 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
2 metidss 33525 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
3 dmss 5909 . . . . . . . . 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 5936 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
64, 5sseqtrdi 4032 . . . . . . 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 5698 . . . . . . . . . 10 (𝑋 Γ— 𝑋) βŠ† (V Γ— V)
92, 8sstrdi 3994 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (V Γ— V))
10 df-rel 5689 . . . . . . . . 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 5950 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
1512, 13, 14syl2anc 582 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
167, 15sseldd 3983 . . . . 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 5950 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
1912, 17, 18syl2anc 582 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
207, 19sseldd 3983 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ 𝑋)
21 psmetsym 24236 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 24232 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„*)
2423fovcdmda 7598 . . . . 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 2829 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5945 . . . . . . . 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 6182 . . . . . . 7 ran (𝑋 Γ— 𝑋) = 𝑋
3028, 29sseqtrdi 4032 . . . . . 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 5951 . . . . . 6 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3312, 13, 32syl2anc 582 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3431, 33sseldd 3983 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ 𝑋)
3523fovcdmda 7598 . . . 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 5951 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3812, 17, 37syl2anc 582 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3931, 38sseldd 3983 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ 𝑋)
40 psmetsym 24236 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
411, 39, 34, 40syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
4223fovcdmda 7598 . . . . 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 2830 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ∈ ℝ*)
45 psmettri2 24235 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
47 psmetsym 24236 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
481, 16, 34, 47syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
4916, 34jca 510 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
50 metidv 33526 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴(~Metβ€˜π·)𝐡 ↔ (𝐴𝐷𝐡) = 0))
5150biimpa 475 . . . . . . . 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 2770 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐴) = 0)
5453oveq1d 7441 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (0 +𝑒 (𝐡𝐷𝐸)))
55 xaddlid 13261 . . . . . 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 2768 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5846, 57breqtrd 5178 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐸))
59 psmettri2 24235 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 24236 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 510 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
64 metidv 33526 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐸(~Metβ€˜π·)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 475 . . . . . . . 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 2768 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = 0)
6867oveq2d 7442 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐡) +𝑒 0))
69 xaddrid 13260 . . . . . 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 2772 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = (𝐡𝐷𝐹))
7260, 71breqtrd 5178 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ (𝐡𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 13181 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹))
7423fovcdmda 7598 . . . 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 24235 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 7441 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddlid 13261 . . . . . 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 2768 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 5178 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐹))
83 psmettri2 24235 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddrid 13260 . . . . . 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 7442 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2778 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 5178 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 13181 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))
91 xrletri3 13173 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐡𝐷𝐹) ∈ ℝ*) β†’ ((𝐴𝐷𝐸) = (𝐡𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹) ∧ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 582 . 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949   class class class wbr 5152   Γ— cxp 5680  dom cdm 5682  ran crn 5683  Rel wrel 5687  β€˜cfv 6553  (class class class)co 7426  0cc0 11146  β„*cxr 11285   ≀ cle 11287   +𝑒 cxad 13130  PsMetcpsmet 21270  ~Metcmetid 33520
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-xadd 13133  df-psmet 21278  df-metid 33522
This theorem is referenced by:  pstmfval  33530
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