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Theorem zrtermoringc 20607
Description: The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
zrtermoringc.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
zrtermoringc.c 𝐢 = (RingCatβ€˜π‘ˆ)
zrtermoringc.z (πœ‘ β†’ 𝑍 ∈ (Ring βˆ– NzRing))
zrtermoringc.e (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
Assertion
Ref Expression
zrtermoringc (πœ‘ β†’ 𝑍 ∈ (TermOβ€˜πΆ))

Proof of Theorem zrtermoringc
Dummy variables π‘Ž β„Ž π‘Ÿ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zrtermoringc.c . . . . . . . . . 10 𝐢 = (RingCatβ€˜π‘ˆ)
2 eqid 2725 . . . . . . . . . 10 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
3 zrtermoringc.u . . . . . . . . . 10 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
41, 2, 3ringcbas 20582 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜πΆ) = (π‘ˆ ∩ Ring))
54eleq2d 2811 . . . . . . . 8 (πœ‘ β†’ (π‘Ÿ ∈ (Baseβ€˜πΆ) ↔ π‘Ÿ ∈ (π‘ˆ ∩ Ring)))
6 elin 3957 . . . . . . . . 9 (π‘Ÿ ∈ (π‘ˆ ∩ Ring) ↔ (π‘Ÿ ∈ π‘ˆ ∧ π‘Ÿ ∈ Ring))
76simprbi 495 . . . . . . . 8 (π‘Ÿ ∈ (π‘ˆ ∩ Ring) β†’ π‘Ÿ ∈ Ring)
85, 7biimtrdi 252 . . . . . . 7 (πœ‘ β†’ (π‘Ÿ ∈ (Baseβ€˜πΆ) β†’ π‘Ÿ ∈ Ring))
98imp 405 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ π‘Ÿ ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (Ring βˆ– NzRing))
1110adantr 479 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ 𝑍 ∈ (Ring βˆ– NzRing))
12 eqid 2725 . . . . . . 7 (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘Ÿ)
13 eqid 2725 . . . . . . 7 (0gβ€˜π‘) = (0gβ€˜π‘)
14 eqid 2725 . . . . . . 7 (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))
1512, 13, 14c0rhm 20470 . . . . . 6 ((π‘Ÿ ∈ Ring ∧ 𝑍 ∈ (Ring βˆ– NzRing)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍))
169, 11, 15syl2anc 582 . . . . 5 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍))
17 simpr 483 . . . . . 6 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍))
183adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ π‘ˆ ∈ 𝑉)
19 eqid 2725 . . . . . . . . . 10 (Hom β€˜πΆ) = (Hom β€˜πΆ)
20 simpr 483 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ π‘Ÿ ∈ (Baseβ€˜πΆ))
21 zrtermoringc.e . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
2210eldifad 3953 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑍 ∈ Ring)
2321, 22elind 4189 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑍 ∈ (π‘ˆ ∩ Ring))
2423, 4eleqtrrd 2828 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ (Baseβ€˜πΆ))
2524adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ 𝑍 ∈ (Baseβ€˜πΆ))
261, 2, 18, 19, 20, 25ringchom 20584 . . . . . . . . 9 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘Ÿ(Hom β€˜πΆ)𝑍) = (π‘Ÿ RingHom 𝑍))
2726eqcomd 2731 . . . . . . . 8 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘Ÿ RingHom 𝑍) = (π‘Ÿ(Hom β€˜πΆ)𝑍))
2827eleq2d 2811 . . . . . . 7 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍) ↔ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍)))
2928biimpa 475 . . . . . 6 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍))
3026eleq2d 2811 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) ↔ β„Ž ∈ (π‘Ÿ RingHom 𝑍)))
31 eqid 2725 . . . . . . . . . . 11 (Baseβ€˜π‘) = (Baseβ€˜π‘)
3212, 31rhmf 20423 . . . . . . . . . 10 (β„Ž ∈ (π‘Ÿ RingHom 𝑍) β†’ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘))
3330, 32biimtrdi 252 . . . . . . . . 9 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)))
3433adantr 479 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)))
35 ffn 6717 . . . . . . . . . . 11 (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) β†’ β„Ž Fn (Baseβ€˜π‘Ÿ))
3635adantl 480 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) β†’ β„Ž Fn (Baseβ€˜π‘Ÿ))
37 fvex 6903 . . . . . . . . . . . 12 (0gβ€˜π‘) ∈ V
3837, 14fnmpti 6693 . . . . . . . . . . 11 (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) Fn (Baseβ€˜π‘Ÿ)
3938a1i 11 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) Fn (Baseβ€˜π‘Ÿ))
4031, 130ringbas 20464 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring βˆ– NzRing) β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4241adantr 479 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4342feq3d 6704 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) ↔ β„Ž:(Baseβ€˜π‘Ÿ)⟢{(0gβ€˜π‘)}))
44 fvconst 7167 . . . . . . . . . . . . . . 15 ((β„Ž:(Baseβ€˜π‘Ÿ)⟢{(0gβ€˜π‘)} ∧ π‘Ž ∈ (Baseβ€˜π‘Ÿ)) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘))
4544ex 411 . . . . . . . . . . . . . 14 (β„Ž:(Baseβ€˜π‘Ÿ)⟢{(0gβ€˜π‘)} β†’ (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘)))
4643, 45biimtrdi 252 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) β†’ (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘))))
4746adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) β†’ (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘))))
4847imp31 416 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜π‘Ÿ)) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘))
49 eqidd 2726 . . . . . . . . . . . . 13 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))
50 eqidd 2726 . . . . . . . . . . . . 13 ((π‘Ž ∈ (Baseβ€˜π‘Ÿ) ∧ π‘₯ = π‘Ž) β†’ (0gβ€˜π‘) = (0gβ€˜π‘))
51 id 22 . . . . . . . . . . . . 13 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ π‘Ž ∈ (Baseβ€˜π‘Ÿ))
5237a1i 11 . . . . . . . . . . . . 13 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (0gβ€˜π‘) ∈ V)
5349, 50, 51, 52fvmptd 7005 . . . . . . . . . . . 12 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))β€˜π‘Ž) = (0gβ€˜π‘))
5453adantl 480 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜π‘Ÿ)) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))β€˜π‘Ž) = (0gβ€˜π‘))
5548, 54eqtr4d 2768 . . . . . . . . . 10 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜π‘Ÿ)) β†’ (β„Žβ€˜π‘Ž) = ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))β€˜π‘Ž))
5636, 39, 55eqfnfvd 7036 . . . . . . . . 9 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) ∧ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘)) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))
5756ex 411 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))))
5834, 57syld 47 . . . . . . 7 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ (β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))))
5958alrimiv 1922 . . . . . 6 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ βˆ€β„Ž(β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))))
6017, 29, 593jca 1125 . . . . 5 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍)) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) ∧ βˆ€β„Ž(β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))))
6116, 60mpdan 685 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) ∧ βˆ€β„Ž(β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))))
62 eleq1 2813 . . . . 5 (β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) β†’ (β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) ↔ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍)))
6362eqeu 3695 . . . 4 (((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍) ∧ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) ∧ βˆ€β„Ž(β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍) β†’ β„Ž = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))) β†’ βˆƒ!β„Ž β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍))
6461, 63syl 17 . . 3 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ βˆƒ!β„Ž β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍))
6564ralrimiva 3136 . 2 (πœ‘ β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜πΆ)βˆƒ!β„Ž β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍))
661ringccat 20595 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
673, 66syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
682, 19, 67, 24istermo 17980 . 2 (πœ‘ β†’ (𝑍 ∈ (TermOβ€˜πΆ) ↔ βˆ€π‘Ÿ ∈ (Baseβ€˜πΆ)βˆƒ!β„Ž β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍)))
6965, 68mpbird 256 1 (πœ‘ β†’ 𝑍 ∈ (TermOβ€˜πΆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆƒ!weu 2556  βˆ€wral 3051  Vcvv 3463   βˆ– cdif 3938   ∩ cin 3940  {csn 4625   ↦ cmpt 5227   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413  Basecbs 17174  Hom chom 17238  0gc0g 17415  Catccat 17638  TermOctermo 17965  Ringcrg 20172   RingHom crh 20407  NzRingcnzr 20450  RingCatcringc 20577
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9919  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-xnn0 12570  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-hash 14317  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-hom 17251  df-cco 17252  df-0g 17417  df-cat 17642  df-cid 17643  df-homf 17644  df-ssc 17787  df-resc 17788  df-subc 17789  df-termo 17968  df-estrc 18107  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-grp 18892  df-minusg 18893  df-ghm 19167  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-rhm 20410  df-nzr 20451  df-ringc 20578
This theorem is referenced by:  nzerooringczr  21405
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