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Theorem prlngsymquadlem 29191

Description: Lemma for prlngsymquad 29192. (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)
prlngsymquad.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
prlngsymquad.3	⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
prlngsymquad.4	⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
prlngsymquadlem.t	⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)

**Assertion**
Ref	Expression
prlngsymquadlem	⊢ (𝜑 → 𝑇 = 𝑊)

**Proof of Theorem prlngsymquadlem**
Step	Hyp	Ref	Expression
1		symquadprlng.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2763	. 2 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		symquadprlng.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		symquadprlng.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		symquadprlng.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
6		symquadprlng.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		symquadprlng.r	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
8		prlngsymquad.4	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
9	3, 7, 4, 8	prlngrcl2 29171	. . . . 5 ⊢ (𝜑 → (𝑊𝐿𝑋) ∈ ran 𝐿)
10	1, 2, 3, 4, 5, 6, 9	tglnne 28879	. . . 4 ⊢ (𝜑 → 𝑊 ≠ 𝑋)
11	1, 2, 3, 4, 5, 6, 10	tglinecom 28886	. . 3 ⊢ (𝜑 → (𝑊𝐿𝑋) = (𝑋𝐿𝑊))
12	11, 9	eqeltrrd 2864	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ∈ ran 𝐿)
13		prlngsymquad.3	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
14	3, 7, 4, 13	prlngrcl2 29171	. 2 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
15		symquadprlng.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16		symquadprlng.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
17		prlngsymquad.2	. . . 4 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
18	1, 2, 3, 4, 6, 15, 16, 5, 17	tglineneq 28896	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
19	4	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiG)
20	13	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
21		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
22	3, 7, 19, 20, 21	prlngin0 29172	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
23	1, 2, 3, 4, 6, 15, 16, 17	ncolne1 28876	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
24	1, 2, 3, 4, 6, 15, 23	tglinerflx1 28884	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑌))
25	24	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑌))
26	10	necomd 3013	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝑊)
27	1, 2, 3, 4, 6, 5, 26	tglinerflx1 28884	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑊))
28	27	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑊))
29		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑊) = (𝑍𝐿𝑊))
30	28, 29	eleqtrd 2865	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑍𝐿𝑊))
31	25, 30	elind 4154	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)))
32	31	ne0d 4296	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) ≠ ∅)
33	32	neneqd 2963	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ¬ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
34	22, 33	pm2.65da 828	. . . 4 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → ¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
35		nne 2962	. . . 4 ⊢ (¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊) ↔ (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
36	34, 35	sylib 221	. . 3 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
37	18, 36	mteqand 3049	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ≠ (𝑍𝐿𝑊))
38		prlngsymquadlem.t	. . . . . 6 ⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)
39		symquadprlng.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
40		eqid 2763	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
41	1, 3, 2, 4, 15, 16, 6, 17	ncoltgdim2 28812	. . . . . . . 8 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
42	1, 39, 2, 4, 41, 6, 16	midcl 29064	. . . . . . 7 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) ∈ 𝑃)
43		eqid 2763	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍)) = ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))
44	1, 39, 2, 3, 40, 4, 42, 43, 15	mircl 28916	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ∈ 𝑃)
45	38, 44	eqeltrid 2867	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
46	3, 7, 4, 8	prlngrcl1 29170	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
47	1, 2, 3, 4, 15, 16, 46	tglnne 28879	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
48	47	necomd 3013	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
49	1, 40, 43, 4, 42, 16, 15	mirleqb 28949	. . . . . . . 8 ⊢ (𝜑 → (𝑍 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
50	49	necon3bid 3002	. . . . . . 7 ⊢ (𝜑 → (𝑍 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
51	48, 50	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
52		eqidd 2764	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍))
53	1, 39, 2, 4, 41, 6, 16, 40, 42	ismidb 29065	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ↔ (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍)))
54	52, 53	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋))
55	54	eqcomd 2769	. . . . . . 7 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = 𝑍)
56	1, 39, 2, 3, 40, 4, 42, 43, 6, 55	mircom 28918	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = 𝑋)
57	38	eqcomi 2772	. . . . . . 7 ⊢ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇)
59	51, 56, 58	3netr3d 3034	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑇)
60	1, 2, 3, 4, 6, 45, 59	tglinerflx2 28885	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑇))
61	1, 40, 43, 4, 42, 6, 15	mirleqb 28949	. . . . . . . 8 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
62	61	necon3bid 3002	. . . . . . 7 ⊢ (𝜑 → (𝑋 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
63	23, 62	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
64	63, 55, 58	3netr3d 3034	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ 𝑇)
65	1, 2, 3, 4, 16, 45, 64	tglinerflx2 28885	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑍𝐿𝑇))
66	60, 65	elind 4154	. . 3 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)))
67		symquadprlng.1	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
68	1, 2, 3, 4, 15, 16, 47	tglinecom 28886	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
69		eqid 2763	. . . . . . 7 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
70		eqid 2763	. . . . . . 7 ⊢ (midG‘𝐺) = (midG‘𝐺)
71	1, 3, 2, 4, 15, 16, 6, 17	ncolcom 28808	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
72	71	orsild 1019	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑍𝐿𝑌))
73	6, 72	eldifd 3917	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
74	1, 39, 2, 4, 41, 16, 6	midcom 29069	. . . . . . . 8 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑋(midG‘𝐺)𝑍))
75	1, 39, 2, 4, 41, 15, 45, 40, 42	ismidb 29065	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ↔ (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍)))
76	38, 75	mpbii 236	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍))
77	76	eqcomd 2769	. . . . . . . 8 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑇))
78	74, 77	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑌(midG‘𝐺)𝑇))
79	1, 3, 69, 7, 70, 4, 67, 16, 15, 73, 45, 78, 48	prlngmid2 29189	. . . . . 6 ⊢ (𝜑 → (𝑍𝐿𝑌) ∥ (𝑋𝐿𝑇))
80	68, 79	eqbrtrd 5134	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑇))
81	8, 11	breqtrd 5138	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑊))
82	1, 2, 3, 4, 6, 45, 59	tglinerflx1 28884	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑇))
83	1, 7, 4, 67, 80, 81, 82, 27	prlngeq 29185	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑇) = (𝑋𝐿𝑊))
84	1, 3, 2, 4, 15, 16, 6, 17	ncolrot2 28810	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
85	84	orsild 1019	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
86	16, 85	eldifd 3917	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
87	1, 3, 69, 7, 70, 4, 67, 6, 15, 86, 45, 77, 23	prlngmid2 29189	. . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑇))
88	1, 2, 3, 4, 16, 45, 64	tglinerflx1 28884	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑇))
89	1, 2, 3, 4, 16, 5, 14	tglnne 28879	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
90	1, 2, 3, 4, 16, 5, 89	tglinerflx1 28884	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
91	1, 7, 4, 67, 87, 13, 88, 90	prlngeq 29185	. . . 4 ⊢ (𝜑 → (𝑍𝐿𝑇) = (𝑍𝐿𝑊))
92	83, 91	ineq12d 4175	. . 3 ⊢ (𝜑 → ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)) = ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
93	66, 92	eleqtrd 2865	. 2 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
94	1, 2, 3, 4, 6, 5, 26	tglinerflx2 28885	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑋𝐿𝑊))
95	1, 2, 3, 4, 16, 5, 89	tglinerflx2 28885	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑍𝐿𝑊))
96	94, 95	elind 4154	. 2 ⊢ (𝜑 → 𝑊 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
97	1, 2, 3, 4, 12, 14, 37, 93, 96	tglineineq 28894	1 ⊢ (𝜑 → 𝑇 = 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∩ cin 3905 ∅c0 4287 class class class wbr 5110 ran crn 5664 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 distcds 17320 TarskiGcstrkg 28674 TarskiG_Ecstrkge 28679 Itvcitv 28680 LineGclng 28681 pInvGcmir 28907 hlGcplng 29033 midGcmid 29059 parlnGcprlng 29164

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-fz 13537 df-fzo 13685 df-hash 14369 df-word 14553 df-concat 14610 df-s1 14636 df-s2 14887 df-s3 14888 df-trkgc 28695 df-trkgb 28696 df-trkgcb 28697 df-trkge 28698 df-trkgld 28699 df-trkg 28700 df-cgrg 28758 df-ismt 28780 df-leg 28830 df-hlg 28848 df-mir 28908 df-rag 28952 df-perpg 28954 df-hpg 29018 df-plng 29034 df-mid 29061 df-lmi 29062 df-cgra 29097 df-prlng 29165

This theorem is referenced by: prlngsymquad 29192 prlngsymquadopp 29193

W3C validator