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Mirrors > Home > MPE Home > Th. List > tgidinside | Structured version Visualization version GIF version |
Description: Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
Ref | Expression |
---|---|
tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
lnxfr.r | ⊢ ∼ = (cgrG‘𝐺) |
lnxfr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
lnxfr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
lnxfr.d | ⊢ − = (dist‘𝐺) |
tgidinside.1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) |
tgidinside.2 | ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
tgidinside.3 | ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) |
Ref | Expression |
---|---|
tgidinside | ⊢ (𝜑 → 𝑍 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | lnxfr.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tglngval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃) |
8 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃) |
10 | tgidinside.1 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) | |
11 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑌)) |
12 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
13 | 12 | oveq2d 7161 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑋) = (𝑋𝐼𝑌)) |
14 | 11, 13 | eleqtrrd 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑋)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 26179 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑍) |
16 | lnxfr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐴 ∈ 𝑃) |
18 | tgidinside.2 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) | |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
20 | 1, 2, 3, 5, 7, 9, 7, 17, 19, 15 | tgcgreq 26195 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴) |
21 | 15, 20 | eqtr3d 2855 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 = 𝐴) |
22 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
23 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG) |
24 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃) |
25 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
26 | 25 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃) |
27 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃) |
28 | lnxfr.r | . . 3 ⊢ ∼ = (cgrG‘𝐺) | |
29 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑃) |
30 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
31 | 30 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐵 ∈ 𝑃) |
32 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
33 | 1, 22, 3, 4, 6, 8, 25, 10 | btwncolg3 26270 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
34 | 33 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
35 | 18 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
36 | tgidinside.3 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) | |
37 | 36 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 − 𝑍) = (𝑌 − 𝐴)) |
38 | 1, 22, 3, 23, 24, 26, 27, 28, 29, 31, 2, 32, 34, 35, 37 | lnid 26283 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 = 𝐴) |
39 | 21, 38 | pm2.61dane 3101 | 1 ⊢ (𝜑 → 𝑍 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 cgrGccgrg 26223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 df-cgrg 26224 |
This theorem is referenced by: miduniq 26398 ragflat2 26416 |
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