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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > transportprops | Structured version Visualization version GIF version |
Description: Calculate the defining properties of the transport function. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
transportprops | ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvtransport 32736 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉))) | |
2 | 1 | eqcomd 2784 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)) |
3 | transportcl 32737 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) ∈ (𝔼‘𝑁)) | |
4 | segconeu 32715 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → ∃!𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) | |
5 | opeq2 4639 | . . . . . 6 ⊢ (𝑟 = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) → 〈𝐶, 𝑟〉 = 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉) | |
6 | 5 | breq2d 4900 | . . . . 5 ⊢ (𝑟 = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) → (𝐷 Btwn 〈𝐶, 𝑟〉 ↔ 𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉)) |
7 | opeq2 4639 | . . . . . 6 ⊢ (𝑟 = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) → 〈𝐷, 𝑟〉 = 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉) | |
8 | 7 | breq1d 4898 | . . . . 5 ⊢ (𝑟 = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) → (〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉 ↔ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉)) |
9 | 6, 8 | anbi12d 624 | . . . 4 ⊢ (𝑟 = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) → ((𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉) ↔ (𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉))) |
10 | 9 | riota2 6907 | . . 3 ⊢ (((〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) ∈ (𝔼‘𝑁) ∧ ∃!𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) → ((𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉) ↔ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉))) |
11 | 3, 4, 10 | syl2anc 579 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → ((𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉) ↔ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) = (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉))) |
12 | 2, 11 | mpbird 249 | 1 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃!wreu 3092 〈cop 4404 class class class wbr 4888 ‘cfv 6137 ℩crio 6884 (class class class)co 6924 ℕcn 11379 𝔼cee 26254 Btwn cbtwn 26255 Cgrccgr 26256 TransportToctransport 32733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-ee 26257 df-btwn 26258 df-cgr 26259 df-ofs 32687 df-transport 32734 |
This theorem is referenced by: (None) |
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