mapdpglem32 - Mathbox for Norm Megill

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Theorem mapdpglem32 41233

Description: Lemma for mapdpg 41234. Uniqueness part - consolidate hypotheses in mapdpglem31 41231. (Contributed by NM, 23-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpg.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpg.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpg.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpg.v	⊢ 𝑉 = (Base‘𝑈)
mapdpg.s	⊢ − = (-_g‘𝑈)
mapdpg.z	⊢ 0 = (0_g‘𝑈)
mapdpg.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpg.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpg.f	⊢ 𝐹 = (Base‘𝐶)
mapdpg.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpg.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpg.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpg.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdpg.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdpg.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpg.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpg.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))

**Assertion**
Ref	Expression
mapdpglem32	⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖)

Distinct variable groups: 𝐶,ℎ ℎ,𝐹 ℎ,𝐺 ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ − ,ℎ 𝑈,ℎ ℎ,𝑋 ℎ,𝑌 ℎ,𝑖 Allowed substitution hints: 𝜑(ℎ,𝑖) 𝐶(𝑖) 𝑅(𝑖) 𝑈(𝑖) 𝐹(𝑖) 𝐺(𝑖) 𝐻(ℎ,𝑖) 𝐽(𝑖) 𝐾(ℎ,𝑖) 𝑀(𝑖) − (𝑖) 𝑁(𝑖) 𝑉(ℎ,𝑖) 𝑊(ℎ,𝑖) 𝑋(𝑖) 𝑌(𝑖) 0 (ℎ,𝑖)

Proof of Theorem mapdpglem32 Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdpg.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpg.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdpg.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdpg.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		mapdpg.s	. . . 4 ⊢ − = (-_g‘𝑈)
6		mapdpg.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
7		mapdpg.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
8		mapdpg.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		mapdpg.f	. . . 4 ⊢ 𝐹 = (Base‘𝐶)
10		mapdpg.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
11		mapdpg.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdpg.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13	12	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		mapdpg.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
15	14	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
16		mapdpg.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
17	16	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
18		mapdpg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
19	18	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝐺 ∈ 𝐹)
20		mapdpg.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
21	20	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
22		mapdpg.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
23	22	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
24		simp2l 1196	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ ∈ 𝐹)
25		simp3l 1198	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))
26	24, 25	jca 510	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
27		simp2r 1197	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑖 ∈ 𝐹)
28		simp3r 1199	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
29	27, 28	jca 510	. . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
30		eqid 2725	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
31		eqid 2725	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
32		eqid 2725	. . . 4 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
33		eqid 2725	. . . 4 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33	mapdpglem26 41226	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33	mapdpglem27 41227	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})(𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))
36		reeanv 3217	. . 3 ⊢ (∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖))) ↔ (∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ ∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})(𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖))))
37	34, 35, 36	sylanbrc 581	. 2 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖))))
38	13	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
39	15	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
40	17	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
41	19	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝐺 ∈ 𝐹)
42	21	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
43	23	3ad2ant1 1130	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
44		simp12l 1283	. . . . . 6 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → ℎ ∈ 𝐹)
45		simp13l 1285	. . . . . 6 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))
46	44, 45	jca 510	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
47		simp12r 1284	. . . . . 6 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝑖 ∈ 𝐹)
48		simp13r 1286	. . . . . 6 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
49	47, 48	jca 510	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
50		eldifi 4119	. . . . . . 7 ⊢ (𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) → 𝑣 ∈ (Base‘(Scalar‘𝑈)))
51	50	adantl 480	. . . . . 6 ⊢ ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) → 𝑣 ∈ (Base‘(Scalar‘𝑈)))
52	51	3ad2ant2 1131	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝑣 ∈ (Base‘(Scalar‘𝑈)))
53		simp3l 1198	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖))
54		simp3r 1199	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))
55		eldifi 4119	. . . . . . 7 ⊢ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) → 𝑢 ∈ (Base‘(Scalar‘𝑈)))
56	55	adantr 479	. . . . . 6 ⊢ ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) → 𝑢 ∈ (Base‘(Scalar‘𝑈)))
57	56	3ad2ant2 1131	. . . . 5 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → 𝑢 ∈ (Base‘(Scalar‘𝑈)))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 46, 49, 30, 31, 32, 33, 52, 53, 54, 57	mapdpglem31 41231	. . . 4 ⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖)))) → ℎ = 𝑖)
59	58	3exp 1116	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) → ((ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖))) → ℎ = 𝑖)))
60	59	rexlimdvv 3201	. 2 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·_𝑠 ‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·_𝑠 ‘𝐶)(𝐺𝑅𝑖))) → ℎ = 𝑖))
61	37, 60	mpd 15	1 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 ∖ cdif 3937 {csn 4624 ‘cfv 6542 (class class class)co 7415 Basecbs 17177 Scalarcsca 17233 ·_𝑠 cvsca 17234 0_gc0g 17418 -_gcsg 18894 LSpanclspn 20857 HLchlt 38877 LHypclh 39512 DVecHcdvh 40606 LCDualclcd 41114 mapdcmpd 41152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-mre 17563 df-mrc 17564 df-acs 17566 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-oppg 19299 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-lsatoms 38503 df-lshyp 38504 df-lcv 38546 df-lfl 38585 df-lkr 38613 df-ldual 38651 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tgrp 40271 df-tendo 40283 df-edring 40285 df-dveca 40531 df-disoa 40557 df-dvech 40607 df-dib 40667 df-dic 40701 df-dih 40757 df-doch 40876 df-djh 40923 df-lcdual 41115 df-mapd 41153

This theorem is referenced by: mapdpg 41234

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