mapdpglem32 - Mathbox for Norm Megill

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Theorem mapdpglem32 40576

Description: Lemma for mapdpg 40577. Uniqueness part - consolidate hypotheses in mapdpglem31 40574. (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 1134	. . . 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 1134	. . . 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 1134	. . . 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 1134	. . . 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 1134	. . . 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 1134	. . . 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 1200	. . . . 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 1202	. . . . 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 513	. . . 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 1201	. . . . 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 1203	. . . . 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 513	. . . 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 2733	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
31		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
32		eqid 2733	. . . 4 ⊢ ( ·_𝑠 ‘𝐶) = ( ·_𝑠 ‘𝐶)
33		eqid 2733	. . . 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 40569	. . 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 40570	. . 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 3227	. . 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 584	. 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 1134	. . . . 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 1134	. . . . 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 1134	. . . . 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 1134	. . . . 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 1134	. . . . 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 1134	. . . . 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 1287	. . . . . 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 1289	. . . . . 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 513	. . . . 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 1288	. . . . . 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 1290	. . . . . 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 513	. . . . 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 4127	. . . . . . 7 ⊢ (𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) → 𝑣 ∈ (Base‘(Scalar‘𝑈)))
51	50	adantl 483	. . . . . 6 ⊢ ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) → 𝑣 ∈ (Base‘(Scalar‘𝑈)))
52	51	3ad2ant2 1135	. . . . 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 1202	. . . . 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 1203	. . . . 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 4127	. . . . . . 7 ⊢ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) → 𝑢 ∈ (Base‘(Scalar‘𝑈)))
56	55	adantr 482	. . . . . 6 ⊢ ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖ {(0_g‘(Scalar‘𝑈))})) → 𝑢 ∈ (Base‘(Scalar‘𝑈)))
57	56	3ad2ant2 1135	. . . . 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 40574	. . . 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 1120	. . 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 3211	. 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∖ cdif 3946 {csn 4629 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 ·_𝑠 cvsca 17201 0_gc0g 17385 -_gcsg 18821 LSpanclspn 20582 HLchlt 38220 LHypclh 38855 DVecHcdvh 39949 LCDualclcd 40457 mapdcmpd 40495

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-riotaBAD 37823

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-undef 8258 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-0g 17387 df-mre 17530 df-mrc 17531 df-acs 17533 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-cntz 19181 df-oppg 19210 df-lsm 19504 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-drng 20359 df-lmod 20473 df-lss 20543 df-lsp 20583 df-lvec 20714 df-lsatoms 37846 df-lshyp 37847 df-lcv 37889 df-lfl 37928 df-lkr 37956 df-ldual 37994 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 df-tgrp 39614 df-tendo 39626 df-edring 39628 df-dveca 39874 df-disoa 39900 df-dvech 39950 df-dib 40010 df-dic 40044 df-dih 40100 df-doch 40219 df-djh 40266 df-lcdual 40458 df-mapd 40496

This theorem is referenced by: mapdpg 40577

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