mideulem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mideulem		Structured version Visualization version GIF version

Theorem mideulem 27125

Description: Lemma for mideu 27127. We can assume mideulem.9 "without loss of generality". (Contributed by Thierry Arnoux, 25-Nov-2019.)

**Hypotheses**
Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mideu.s	⊢ 𝑆 = (pInvG‘𝐺)
mideu.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mideu.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mideulem.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
mideulem.2	⊢ (𝜑 → 𝑄 ∈ 𝑃)
mideulem.3	⊢ (𝜑 → 𝑂 ∈ 𝑃)
mideulem.4	⊢ (𝜑 → 𝑇 ∈ 𝑃)
mideulem.5	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
mideulem.6	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
mideulem.7	⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
mideulem.8	⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
mideulem.9	⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄))

**Assertion**
Ref	Expression
mideulem	⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Distinct variable groups: 𝑥, − 𝑥,𝐴 𝑥,𝐵 𝑥,𝐼 𝑥,𝑂 𝑥,𝑃 𝑥,𝑄 𝑥,𝑇 𝜑,𝑥 Allowed substitution hints: 𝑆(𝑥) 𝐺(𝑥) 𝐿(𝑥)

Proof of Theorem mideulem Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprrl 777	. . 3 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) ∧ (𝑥 ∈ 𝑃 ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
2		colperpex.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		colperpex.d	. . . 4 ⊢ − = (dist‘𝐺)
4		colperpex.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		colperpex.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
6		colperpex.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐺 ∈ TarskiG)
8		mideu.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
9		mideu.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10	9	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ∈ 𝑃)
11		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐵 ∈ 𝑃)
13		mideulem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
14	13	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ≠ 𝐵)
15		mideulem.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
16	15	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑄 ∈ 𝑃)
17		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
18	17	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑂 ∈ 𝑃)
19		mideulem.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
20	19	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ 𝑃)
21		mideulem.5	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
22	21	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
23		mideulem.6	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
24	23	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
25		mideulem.7	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
26	25	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝐴𝐿𝐵))
27		mideulem.8	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
28	27	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝑄𝐼𝑂))
29		simplr 765	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ 𝑃)
30		simprl 767	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ (𝐵𝐼𝑄))
31		simprr 769	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴 − 𝑂) = (𝐵 − 𝑟))
32	2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 31	opphllem 27124	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))
33	1, 32	reximddv 3162	. 2 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
34		mideulem.9	. . 3 ⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄))
35		eqid 2733	. . . 4 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
36	2, 3, 4, 35, 6, 9, 17, 11, 15	legov 26974	. . 3 ⊢ (𝜑 → ((𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))))
37	34, 36	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟)))
38	33, 37	r19.29a 3153	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ≠ wne 2938 ∃wrex 3068 class class class wbr 5077 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 distcds 16999 TarskiGcstrkg 26816 Itvcitv 26822 LineGclng 26823 ≤Gcleg 26971 pInvGcmir 27041 ⟂Gcperpg 27084

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-oadd 8321 df-er 8518 df-map 8637 df-pm 8638 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-dju 9687 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-xnn0 12334 df-z 12348 df-uz 12611 df-fz 13268 df-fzo 13411 df-hash 14073 df-word 14246 df-concat 14302 df-s1 14329 df-s2 14589 df-s3 14590 df-trkgc 26837 df-trkgb 26838 df-trkgcb 26839 df-trkg 26842 df-cgrg 26900 df-leg 26972 df-mir 27042 df-rag 27083 df-perpg 27085

This theorem is referenced by: midex 27126

W3C validator