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Theorem symquadprlng 29327

Description: Symmetrical quadrilaterals are parallelograms. Theorem 12.18 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)

**Hypotheses**
Ref	Expression
symquadprlng.p	⊢ 𝑃 = (Base‘𝐺)
symquadprlng.d	⊢ − = (dist‘𝐺)
symquadprlng.l	⊢ 𝐿 = (LineG‘𝐺)
symquadprlng.r	⊢ ∥ = (parlnG‘𝐺)
symquadprlng.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadprlng.1	⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
symquadprlng.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadprlng.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
symquadprlng.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
symquadprlng.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
symquadprlng.2	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
symquadprlng.3	⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
symquadprlng.4	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
symquadprlng.5	⊢ (𝜑 → 𝑌 ≠ 𝑊)
symquadprlng.6	⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
symquadprlng.7	⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))

**Assertion**
Ref	Expression
symquadprlng	⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))

**Proof of Theorem symquadprlng**
Step	Hyp	Ref	Expression
1		symquadprlng.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		symquadprlng.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
3		eqid 2762	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		symquadprlng.r	. . 3 ⊢ ∥ = (parlnG‘𝐺)
5		eqid 2762	. . 3 ⊢ (midG‘𝐺) = (midG‘𝐺)
6		symquadprlng.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		symquadprlng.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
8		symquadprlng.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9		symquadprlng.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10		symquadprlng.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11		eqid 2762	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
12		symquadprlng.4	. . . . . 6 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
13	1, 2, 11, 6, 9, 10, 8, 12	ncolrot2 28913	. . . . 5 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
14	13	orsild 1019	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
15	10, 14	eldifd 3913	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
16		symquadprlng.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
17		symquadprlng.d	. . . . . 6 ⊢ − = (dist‘𝐺)
18		eqid 2762	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
19		eqid 2762	. . . . . 6 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
20		symquadprlng.5	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
21	1, 11, 2, 6, 9, 16, 20	tgelrnln 28985	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑊) ∈ ran 𝐿)
22		symquadprlng.7	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))
23	1, 2, 11, 6, 21, 22	tglnpt 28899	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
24		symquadprlng.2	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
25		symquadprlng.3	. . . . . 6 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
26		symquadprlng.6	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
27	26	orcd 887	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
28	22	orcd 887	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑌𝐿𝑊) ∨ 𝑌 = 𝑊))
29	1, 17, 11, 2, 18, 6, 19, 8, 9, 10, 16, 23, 12, 20, 24, 25, 27, 28	symquadlem 29048	. . . . 5 ⊢ (𝜑 → 𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍))
30	1, 2, 11, 6, 9, 10, 8, 12	ncoltgdim2 28915	. . . . . 6 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
31	1, 17, 11, 6, 30, 10, 8, 18, 23	ismidb 29170	. . . . 5 ⊢ (𝜑 → (𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍) ↔ (𝑍(midG‘𝐺)𝑋) = 𝑇))
32	29, 31	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = 𝑇)
33	1, 17, 11, 6, 30, 8, 10	midcom 29174	. . . 4 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
34	1, 17, 2, 6, 8, 9, 10, 16, 24, 25, 12, 20, 26, 22	symquadprlnglem 29052	. . . . . 6 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
35	1, 2, 11, 6, 8, 10, 26	tglngne 28900	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
36	35	necomd 3012	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑋)
37	1, 17, 11, 6, 8, 9, 10, 16, 24	tgcgrcomlr 28829	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) = (𝑊 − 𝑍))
38	37	eqcomd 2768	. . . . . 6 ⊢ (𝜑 → (𝑊 − 𝑍) = (𝑌 − 𝑋))
39	1, 17, 11, 6, 9, 10, 16, 8, 25	tgcgrcomlr 28829	. . . . . 6 ⊢ (𝜑 → (𝑍 − 𝑌) = (𝑋 − 𝑊))
40	1, 2, 11, 6, 9, 16, 23, 28	colcom 28908	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑊𝐿𝑌) ∨ 𝑊 = 𝑌))
41	1, 2, 11, 6, 8, 10, 23, 27	colcom 28908	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
42	1, 17, 11, 2, 18, 6, 19, 16, 10, 9, 8, 23, 34, 36, 38, 39, 40, 41	symquadlem 29048	. . . . 5 ⊢ (𝜑 → 𝑊 = (((pInvG‘𝐺)‘𝑇)‘𝑌))
43	1, 17, 11, 6, 30, 9, 16, 18, 23	ismidb 29170	. . . . 5 ⊢ (𝜑 → (𝑊 = (((pInvG‘𝐺)‘𝑇)‘𝑌) ↔ (𝑌(midG‘𝐺)𝑊) = 𝑇))
44	42, 43	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑊) = 𝑇)
45	32, 33, 44	3eqtr4d 2807	. . 3 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
46	1, 11, 2, 6, 8, 9, 10, 12	ncolne1 28980	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 45, 46	prlngmid2 29326	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
48	34	orsild 1019	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑍𝐿𝑌))
49	34	orsird 1020	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 = 𝑌)
50	49	neqned 2964	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
51	1, 11, 2, 6, 10, 9, 50	tglinecom 28990	. . . . 5 ⊢ (𝜑 → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
52	48, 51	neleqtrd 2884	. . . 4 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑌𝐿𝑍))
53	16, 52	eldifd 3913	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))
54	44, 32	eqtr4d 2800	. . 3 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑊) = (𝑍(midG‘𝐺)𝑋))
55	50	necomd 3012	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
56	1, 2, 3, 4, 5, 6, 7, 9, 10, 53, 8, 54, 55	prlngmid2 29326	. 2 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
57	47, 56	jca 521	1 ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 distcds 17357 TarskiGcstrkg 28776 TarskiG_Ecstrkge 28781 Itvcitv 28782 LineGclng 28783 pInvGcmir 29011 hlGcplng 29138 midGcmid 29164 parlnGcprlng 29301

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-dju 9910 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-xnn0 12606 df-z 12620 df-uz 12892 df-fz 13566 df-fzo 13714 df-hash 14399 df-word 14583 df-concat 14640 df-s1 14667 df-s2 14923 df-s3 14924 df-trkgc 28797 df-trkgb 28798 df-trkgcb 28799 df-trkgld 28801 df-trkg 28802 df-cgrg 28861 df-ismt 28883 df-leg 28933 df-hlg 28951 df-mir 29012 df-rag 29056 df-perpg 29058 df-hpg 29123 df-plng 29139 df-mid 29166 df-lmi 29167 df-cgra 29202 df-prlng 29302

This theorem is used by: (None)

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