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Theorem symquadlem 28697
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 28495 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 28563 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1110adantr 480 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7447 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
1413eleq2d 2827 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
1512eqeq2d 2748 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐵 = 𝐴𝐵 = 𝐶))
1614, 15orbi12d 919 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
1711, 16mpbid 232 . . . . . . . 8 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
181, 17mtand 816 . . . . . . 7 (𝜑 → ¬ 𝐴 = 𝐶)
1918neqned 2947 . . . . . 6 (𝜑𝐴𝐶)
2019ad2antrr 726 . . . . 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 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25 symquadlem.m . . . . . 6 𝑀 = (𝑆𝑋)
26 symquadlem.c . . . . . . 7 (𝜑𝐶𝑃)
2726ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝑃)
287ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝑃)
29 symquadlem.x . . . . . . 7 (𝜑𝑋𝑃)
3029ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋𝑃)
31 symquadlem.5 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
3231ad2antrr 726 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
332, 3, 4, 24, 28, 27, 30, 32colcom 28566 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
346ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝑃)
35 symquadlem.d . . . . . . . . 9 (𝜑𝐷𝑃)
3635ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷𝑃)
37 eqid 2737 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
38 simplr 769 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥𝑃)
39 symquadlem.6 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
402, 3, 4, 5, 6, 35, 29, 39colrot2 28568 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
412, 3, 4, 5, 29, 6, 35, 40colcom 28566 . . . . . . . . 9 (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
4241ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43 simpr 484 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44 symquadlem.4 . . . . . . . . 9 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
4544ad2antrr 726 . . . . . . . 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 28488 . . . . . . . . . 10 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
4847ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐴) = (𝐷 𝐶))
4948eqcomd 2743 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐶) = (𝐵 𝐴))
50 symquadlem.2 . . . . . . . . 9 (𝜑𝐵𝐷)
5150ad2antrr 726 . . . . . . . 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 28576 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑥 𝐴))
532, 3, 4, 5, 6, 26, 7, 1ncolcom 28569 . . . . . . . . . 10 (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5453ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5531orcomd 872 . . . . . . . . . . . 12 (𝜑 → (𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5655ord 865 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5718, 56mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐴𝐿𝐶))
5857ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
5918ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
6045eqcomd 2743 . . . . . . . . . . . . . . . . 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 28576 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐶))
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 28488 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 𝑋) = (𝐶 𝑥))
632, 8, 4, 24, 27, 28axtgcgrrflx 28470 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 𝐴) = (𝐴 𝐶))
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 28524 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 28574 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
662, 3, 4, 24, 27, 28, 38, 65colcom 28566 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
6766orcomd 872 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6867ord 865 . . . . . . . . . 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 872 . . . . . . . . . . . 12 (𝜑 → (𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7271ord 865 . . . . . . . . . . 11 (𝜑 → (¬ 𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7370, 72mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝐿𝐷))
7473ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
7570ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
7639ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 28532 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 28574 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
792, 3, 4, 24, 36, 34, 38, 78colcom 28566 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
8079orcomd 872 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8180ord 865 . . . . . . . . . 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 28653 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
8483oveq1d 7446 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐴))
8552, 84eqtr4d 2780 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑋 𝐴))
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 28696 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀𝐶) ∨ 𝐶 = 𝐴))
8786orcomd 872 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8887ord 865 . . 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 28470 . . 3 (𝜑 → (𝐵 𝐷) = (𝐷 𝐵))
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 28575 . 2 (𝜑 → ∃𝑥𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
9289, 91r19.29a 3162 1 (𝜑𝐴 = (𝑀𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  ⟨“cs3 14881  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  cgrGccgrg 28518  pInvGcmir 28660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519  df-mir 28661
This theorem is referenced by:  opphllem  28743
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