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Mirrors > Home > MPE Home > Th. List > hlcgreulem | Structured version Visualization version GIF version |
Description: Lemma for hlcgreu 25513. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hlln.1 | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
hlcgrex.m | ⊢ − = (dist‘𝐺) |
hlcgrex.1 | ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
hlcgrex.2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
hlcgreulem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
hlcgreulem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
hlcgreulem.1 | ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷) |
hlcgreulem.2 | ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷) |
hlcgreulem.3 | ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
hlcgreulem.4 | ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) |
Ref | Expression |
---|---|
hlcgreulem | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | hlcgrex.m | . . 3 ⊢ − = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃) |
8 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃) |
10 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃) |
12 | simplr 792 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃) | |
13 | hlcgreulem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 13 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃) |
15 | hlcgreulem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
16 | 15 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃) |
17 | simprr 796 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦) | |
18 | 17 | necomd 2849 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ≠ 𝐴) |
19 | ishlg.k | . . . . 5 ⊢ 𝐾 = (hlG‘𝐺) | |
20 | hltr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
21 | 20 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷 ∈ 𝑃) |
22 | hlcgreulem.1 | . . . . . . 7 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷) | |
23 | 1, 3, 19, 13, 20, 6, 4, 22 | hlcomd 25499 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋) |
24 | 23 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋) |
25 | simprl 794 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦)) | |
26 | 1, 3, 19, 21, 14, 12, 5, 7, 24, 25 | btwnhl 25509 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦)) |
27 | 1, 2, 3, 5, 14, 7, 12, 26 | tgbtwncom 25383 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑋)) |
28 | hlcgreulem.2 | . . . . . . 7 ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷) | |
29 | 1, 3, 19, 15, 20, 6, 4, 28 | hlcomd 25499 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌) |
30 | 29 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌) |
31 | 1, 3, 19, 21, 16, 12, 5, 7, 30, 25 | btwnhl 25509 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦)) |
32 | 1, 2, 3, 5, 16, 7, 12, 31 | tgbtwncom 25383 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌)) |
33 | hlcgreulem.3 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) | |
34 | 33 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
35 | hlcgreulem.4 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) | |
36 | 35 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑌) = (𝐵 − 𝐶)) |
37 | 1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 27, 32, 34, 36 | tgsegconeq 25381 | . 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌) |
38 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
39 | 1, 38 | eqeltri 2697 | . . . . 5 ⊢ 𝑃 ∈ V |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ V) |
41 | hlcgrex.2 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
42 | 40, 8, 10, 41 | nehash2 13256 | . . 3 ⊢ (𝜑 → 2 ≤ (#‘𝑃)) |
43 | 1, 2, 3, 4, 20, 6, 42 | tgbtwndiff 25401 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) |
44 | 37, 43 | r19.29a 3078 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 hlGchlg 25495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-hlg 25496 |
This theorem is referenced by: hlcgreu 25513 iscgra1 25702 |
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