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Mirrors > Home > MPE Home > Th. List > hlcgreulem | Structured version Visualization version GIF version |
Description: Lemma for hlcgreu 25313. (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 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃) |
8 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃) |
10 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃) |
12 | simplr 788 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃) | |
13 | hlcgreulem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 13 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃) |
15 | hlcgreulem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
16 | 15 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃) |
17 | simprr 792 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦) | |
18 | 17 | necomd 2837 | . . 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 758 | . . . . 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 25299 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋) |
24 | 23 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋) |
25 | simprl 790 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦)) | |
26 | 1, 3, 19, 21, 14, 12, 5, 7, 24, 25 | btwnhl 25309 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦)) |
27 | 1, 2, 3, 5, 14, 7, 12, 26 | tgbtwncom 25183 | . . 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 25299 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌) |
30 | 29 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌) |
31 | 1, 3, 19, 21, 16, 12, 5, 7, 30, 25 | btwnhl 25309 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦)) |
32 | 1, 2, 3, 5, 16, 7, 12, 31 | tgbtwncom 25183 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌)) |
33 | hlcgreulem.3 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) | |
34 | 33 | ad2antrr 758 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
35 | hlcgreulem.4 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) | |
36 | 35 | ad2antrr 758 | . . 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 25181 | . 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌) |
38 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
39 | 1, 38 | eqeltri 2684 | . . . . 5 ⊢ 𝑃 ∈ V |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ V) |
41 | hlcgrex.2 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
42 | 40, 8, 10, 41 | nehash2 13113 | . . 3 ⊢ (𝜑 → 2 ≤ (#‘𝑃)) |
43 | 1, 2, 3, 4, 20, 6, 42 | tgbtwndiff 25201 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) |
44 | 37, 43 | r19.29a 3060 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 hlGchlg 25295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-hlg 25296 |
This theorem is referenced by: hlcgreu 25313 iscgra1 25502 |
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