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Mirrors > Home > MPE Home > Th. List > imasdsf1o | Structured version Visualization version GIF version |
Description: The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
imasdsf1o.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasdsf1o.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasdsf1o.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasdsf1o.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasdsf1o.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
imasdsf1o.d | ⊢ 𝐷 = (dist‘𝑈) |
imasdsf1o.m | ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) |
imasdsf1o.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
imasdsf1o.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
imasdsf1o | ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasdsf1o.u | . 2 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasdsf1o.v | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasdsf1o.f | . 2 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
4 | imasdsf1o.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | imasdsf1o.e | . 2 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
6 | imasdsf1o.d | . 2 ⊢ 𝐷 = (dist‘𝑈) | |
7 | imasdsf1o.m | . 2 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) | |
8 | imasdsf1o.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | imasdsf1o.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
10 | eqid 2610 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
11 | eqid 2610 | . 2 ⊢ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} = {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} | |
12 | eqid 2610 | . 2 ⊢ ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | imasdsf1olem 21988 | 1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 {csn 4125 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ↑𝑚 cmap 7744 1c1 9816 + caddc 9818 -∞cmnf 9951 ℝ*cxr 9952 − cmin 10145 ℕcn 10897 ...cfz 12197 Basecbs 15695 ↾s cress 15696 distcds 15777 Σg cgsu 15924 ℝ*𝑠cxrs 15983 “s cimas 15987 ∞Metcxmt 19552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-gsum 15926 df-xrs 15985 df-imas 15991 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-xmet 19560 |
This theorem is referenced by: imasf1oxmet 21990 imasf1omet 21991 xpsdsval 21996 imasf1obl 22103 |
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