![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lnid | Structured version Visualization version GIF version |
Description: Identity law for points on lines. Theorem 4.18 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) |
lnid.1 | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
lnid.2 | ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
lnid.3 | ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
lnid.4 | ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) |
Ref | Expression |
---|---|
lnid | ⊢ (𝜑 → 𝑍 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | lnxfr.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | tglngval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgcolg.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
6 | lnxfr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | tglngval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | tglngval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
9 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
10 | lnxfr.r | . . . 4 ⊢ ∼ = (cgrG‘𝐺) | |
11 | lnid.1 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
12 | lnid.2 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) | |
13 | lnid.3 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) | |
14 | lnid.4 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) | |
15 | 1, 7, 3, 4, 8, 9, 5, 10, 5, 6, 2, 11, 12, 13, 14 | lncgr 25820 | . . 3 ⊢ (𝜑 → (𝑍 − 𝑍) = (𝑍 − 𝐴)) |
16 | 15 | eqcomd 2805 | . 2 ⊢ (𝜑 → (𝑍 − 𝐴) = (𝑍 − 𝑍)) |
17 | 1, 2, 3, 4, 5, 6, 5, 16 | axtgcgrid 25714 | 1 ⊢ (𝜑 → 𝑍 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 distcds 16276 TarskiGcstrkg 25681 Itvcitv 25687 LineGclng 25688 cgrGccgrg 25761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 df-s1 13616 df-s2 13933 df-s3 13934 df-trkgc 25699 df-trkgb 25700 df-trkgcb 25701 df-trkg 25704 df-cgrg 25762 |
This theorem is referenced by: tgidinside 25822 tgbtwnconn1lem3 25825 |
Copyright terms: Public domain | W3C validator |