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Theorem symquadlem 27252
Description: Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
symquadlem.m 𝑀 = (𝑆𝑋)
symquadlem.a (𝜑𝐴𝑃)
symquadlem.b (𝜑𝐵𝑃)
symquadlem.c (𝜑𝐶𝑃)
symquadlem.d (𝜑𝐷𝑃)
symquadlem.x (𝜑𝑋𝑃)
symquadlem.1 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
symquadlem.2 (𝜑𝐵𝐷)
symquadlem.3 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
symquadlem.4 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
symquadlem.5 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
symquadlem.6 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
Assertion
Ref Expression
symquadlem (𝜑𝐴 = (𝑀𝐶))

Proof of Theorem symquadlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 symquadlem.1 . . . . . . . 8 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
2 mirval.p . . . . . . . . . . 11 𝑃 = (Base‘𝐺)
3 mirval.l . . . . . . . . . . 11 𝐿 = (LineG‘𝐺)
4 mirval.i . . . . . . . . . . 11 𝐼 = (Itv‘𝐺)
5 mirval.g . . . . . . . . . . 11 (𝜑𝐺 ∈ TarskiG)
6 symquadlem.b . . . . . . . . . . 11 (𝜑𝐵𝑃)
7 symquadlem.a . . . . . . . . . . 11 (𝜑𝐴𝑃)
8 mirval.d . . . . . . . . . . . 12 = (dist‘𝐺)
92, 8, 4, 5, 6, 7tgbtwntriv2 27050 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 27118 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1110adantr 481 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12 simpr 485 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7345 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
1413eleq2d 2822 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
1512eqeq2d 2747 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐵 = 𝐴𝐵 = 𝐶))
1614, 15orbi12d 916 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
1711, 16mpbid 231 . . . . . . . 8 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
181, 17mtand 813 . . . . . . 7 (𝜑 → ¬ 𝐴 = 𝐶)
1918neqned 2947 . . . . . 6 (𝜑𝐴𝐶)
2019ad2antrr 723 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝐶)
2120necomd 2996 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝐴)
2221neneqd 2945 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
245ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25 symquadlem.m . . . . . 6 𝑀 = (𝑆𝑋)
26 symquadlem.c . . . . . . 7 (𝜑𝐶𝑃)
2726ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝑃)
287ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝑃)
29 symquadlem.x . . . . . . 7 (𝜑𝑋𝑃)
3029ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋𝑃)
31 symquadlem.5 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
3231ad2antrr 723 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
332, 3, 4, 24, 28, 27, 30, 32colcom 27121 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
346ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝑃)
35 symquadlem.d . . . . . . . . 9 (𝜑𝐷𝑃)
3635ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷𝑃)
37 eqid 2736 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
38 simplr 766 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥𝑃)
39 symquadlem.6 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
402, 3, 4, 5, 6, 35, 29, 39colrot2 27123 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
412, 3, 4, 5, 29, 6, 35, 40colcom 27121 . . . . . . . . 9 (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
4241ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43 simpr 485 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44 symquadlem.4 . . . . . . . . 9 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
4544ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐶) = (𝐷 𝐴))
46 symquadlem.3 . . . . . . . . . . 11 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
472, 8, 4, 5, 7, 6, 26, 35, 46tgcgrcomlr 27043 . . . . . . . . . 10 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
4847ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐴) = (𝐷 𝐶))
4948eqcomd 2742 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐶) = (𝐵 𝐴))
50 symquadlem.2 . . . . . . . . 9 (𝜑𝐵𝐷)
5150ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝐷)
522, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 27, 38, 28, 42, 43, 45, 49, 51tgfscgr 27131 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑥 𝐴))
532, 3, 4, 5, 6, 26, 7, 1ncolcom 27124 . . . . . . . . . 10 (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5453ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5531orcomd 868 . . . . . . . . . . . 12 (𝜑 → (𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5655ord 861 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5718, 56mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐴𝐿𝐶))
5857ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
5918ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
6045eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐴) = (𝐵 𝐶))
612, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 28, 38, 27, 42, 43, 48, 60, 51tgfscgr 27131 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐶))
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 27043 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 𝑋) = (𝐶 𝑥))
632, 8, 4, 24, 27, 28axtgcgrrflx 27025 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 𝐴) = (𝐴 𝐶))
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 27079 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 27129 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
662, 3, 4, 24, 27, 28, 38, 65colcom 27121 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
6766orcomd 868 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6867ord 861 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6959, 68mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐴𝐿𝐶))
7050neneqd 2945 . . . . . . . . . . 11 (𝜑 → ¬ 𝐵 = 𝐷)
7139orcomd 868 . . . . . . . . . . . 12 (𝜑 → (𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7271ord 861 . . . . . . . . . . 11 (𝜑 → (¬ 𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7370, 72mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝐿𝐷))
7473ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
7570ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
7639ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 27087 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 27129 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
792, 3, 4, 24, 36, 34, 38, 78colcom 27121 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
8079orcomd 868 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8180ord 861 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8275, 81mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐵𝐿𝐷))
832, 4, 3, 24, 28, 27, 34, 36, 54, 58, 69, 74, 82tglineinteq 27208 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
8483oveq1d 7344 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐴))
8552, 84eqtr4d 2779 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑋 𝐴))
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 27251 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀𝐶) ∨ 𝐶 = 𝐴))
8786orcomd 868 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8887ord 861 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8922, 88mpd 15 . 2 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 = (𝑀𝐶))
902, 8, 4, 5, 6, 35axtgcgrrflx 27025 . . 3 (𝜑 → (𝐵 𝐷) = (𝐷 𝐵))
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 27130 . 2 (𝜑 → ∃𝑥𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
9289, 91r19.29a 3155 1 (𝜑𝐴 = (𝑀𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5089  cfv 6473  (class class class)co 7329  ⟨“cs3 14646  Basecbs 17001  distcds 17060  TarskiGcstrkg 26990  Itvcitv 26996  LineGclng 26997  cgrGccgrg 27073  pInvGcmir 27215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-oadd 8363  df-er 8561  df-pm 8681  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-dju 9750  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-n0 12327  df-xnn0 12399  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476  df-hash 14138  df-word 14310  df-concat 14366  df-s1 14392  df-s2 14652  df-s3 14653  df-trkgc 27011  df-trkgb 27012  df-trkgcb 27013  df-trkg 27016  df-cgrg 27074  df-mir 27216
This theorem is referenced by:  opphllem  27298
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