![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lncgr | Structured version Visualization version GIF version |
Description: Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (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) |
lncgr.1 | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
lncgr.2 | ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
lncgr.3 | ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑋 − 𝐵)) |
lncgr.4 | ⊢ (𝜑 → (𝑌 − 𝐴) = (𝑌 − 𝐵)) |
Ref | Expression |
---|---|
lncgr | ⊢ (𝜑 → (𝑍 − 𝐴) = (𝑍 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tglngval.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tglngval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tglngval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
6 | tglngval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
7 | tgcolg.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
8 | lnxfr.r | . 2 ⊢ ∼ = (cgrG‘𝐺) | |
9 | lnxfr.d | . 2 ⊢ − = (dist‘𝐺) | |
10 | lnxfr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | lnxfr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | lncgr.2 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) | |
13 | 1, 9, 3, 8, 4, 5, 6, 7 | cgr3id 27459 | . 2 ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝑋𝑌𝑍”〉) |
14 | lncgr.3 | . 2 ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑋 − 𝐵)) | |
15 | lncgr.4 | . 2 ⊢ (𝜑 → (𝑌 − 𝐴) = (𝑌 − 𝐵)) | |
16 | lncgr.1 | . 2 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 6, 9, 10, 7, 11, 12, 13, 14, 15, 16 | tgfscgr 27508 | 1 ⊢ (𝜑 → (𝑍 − 𝐴) = (𝑍 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 distcds 17141 TarskiGcstrkg 27367 Itvcitv 27373 LineGclng 27374 cgrGccgrg 27450 |
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 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
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-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 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8647 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-xnn0 12485 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 df-hash 14230 df-word 14402 df-concat 14458 df-s1 14483 df-s2 14736 df-s3 14737 df-trkgc 27388 df-trkgb 27389 df-trkgcb 27390 df-trkg 27393 df-cgrg 27451 |
This theorem is referenced by: lnid 27510 tgbtwnconn1lem3 27514 krippenlem 27630 midexlem 27632 ragcol 27639 hypcgrlem1 27739 trgcopyeulem 27745 |
Copyright terms: Public domain | W3C validator |