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Mirrors > Home > MPE Home > Th. List > midf | Structured version Visualization version GIF version |
Description: Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
Ref | Expression |
---|---|
midf | ⊢ (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | eqid 2736 | . . . . . 6 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
5 | ismid.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ TarskiG) |
7 | eqid 2736 | . . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
8 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃) | |
9 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃) | |
10 | ismid.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺DimTarskiG≥2) |
12 | 1, 2, 3, 4, 6, 7, 8, 9, 11 | mideu 27680 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ∃!𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) |
13 | 12 | ralrimivva 3197 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃!𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) |
14 | riotacl 7331 | . . . . 5 ⊢ (∃!𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎) → (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) ∈ 𝑃) | |
15 | 14 | 2ralimi 3126 | . . . 4 ⊢ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃!𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎) → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) ∈ 𝑃) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) ∈ 𝑃) |
17 | eqid 2736 | . . . 4 ⊢ (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) | |
18 | 17 | fmpo 8000 | . . 3 ⊢ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)) ∈ 𝑃 ↔ (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))):(𝑃 × 𝑃)⟶𝑃) |
19 | 16, 18 | sylib 217 | . 2 ⊢ (𝜑 → (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))):(𝑃 × 𝑃)⟶𝑃) |
20 | df-mid 27716 | . . . 4 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
21 | fveq2 6842 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
22 | 21, 1 | eqtr4di 2794 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
23 | fveq2 6842 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
24 | 23 | fveq1d 6844 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = ((pInvG‘𝐺)‘𝑚)) |
25 | 24 | fveq1d 6844 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = (((pInvG‘𝐺)‘𝑚)‘𝑎)) |
26 | 25 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) |
27 | 22, 26 | riotaeqbidv 7316 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) |
28 | 22, 22, 27 | mpoeq123dv 7432 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)))) |
29 | 5 | elexd 3465 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
30 | 1 | fvexi 6856 | . . . . . 6 ⊢ 𝑃 ∈ V |
31 | 30, 30 | mpoex 8012 | . . . . 5 ⊢ (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) ∈ V |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))) ∈ V) |
33 | 20, 28, 29, 32 | fvmptd3 6971 | . . 3 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎)))) |
34 | 33 | feq1d 6653 | . 2 ⊢ (𝜑 → ((midG‘𝐺):(𝑃 × 𝑃)⟶𝑃 ↔ (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = (((pInvG‘𝐺)‘𝑚)‘𝑎))):(𝑃 × 𝑃)⟶𝑃)) |
35 | 19, 34 | mpbird 256 | 1 ⊢ (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃!wreu 3351 Vcvv 3445 class class class wbr 5105 × cxp 5631 ⟶wf 6492 ‘cfv 6496 ℩crio 7312 ∈ cmpo 7359 2c2 12208 Basecbs 17083 distcds 17142 TarskiGcstrkg 27369 DimTarskiG≥cstrkgld 27373 Itvcitv 27375 LineGclng 27376 pInvGcmir 27594 midGcmid 27714 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-s3 14738 df-trkgc 27390 df-trkgb 27391 df-trkgcb 27392 df-trkgld 27394 df-trkg 27395 df-cgrg 27453 df-leg 27525 df-mir 27595 df-rag 27636 df-perpg 27638 df-mid 27716 |
This theorem is referenced by: midcl 27719 |
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