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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  zrtermoringc Structured version   Visualization version   GIF version

Theorem zrtermoringc 46442
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 2737 . . . . . . . . . 10 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
3 zrtermoringc.u . . . . . . . . . 10 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
41, 2, 3ringcbas 46383 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜πΆ) = (π‘ˆ ∩ Ring))
54eleq2d 2824 . . . . . . . 8 (πœ‘ β†’ (π‘Ÿ ∈ (Baseβ€˜πΆ) ↔ π‘Ÿ ∈ (π‘ˆ ∩ Ring)))
6 elin 3931 . . . . . . . . 9 (π‘Ÿ ∈ (π‘ˆ ∩ Ring) ↔ (π‘Ÿ ∈ π‘ˆ ∧ π‘Ÿ ∈ Ring))
76simprbi 498 . . . . . . . 8 (π‘Ÿ ∈ (π‘ˆ ∩ Ring) β†’ π‘Ÿ ∈ Ring)
85, 7syl6bi 253 . . . . . . 7 (πœ‘ β†’ (π‘Ÿ ∈ (Baseβ€˜πΆ) β†’ π‘Ÿ ∈ Ring))
98imp 408 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ π‘Ÿ ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (Ring βˆ– NzRing))
1110adantr 482 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ 𝑍 ∈ (Ring βˆ– NzRing))
12 eqid 2737 . . . . . . 7 (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘Ÿ)
13 eqid 2737 . . . . . . 7 (0gβ€˜π‘) = (0gβ€˜π‘)
14 eqid 2737 . . . . . . 7 (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))
1512, 13, 14c0rhm 46284 . . . . . 6 ((π‘Ÿ ∈ Ring ∧ 𝑍 ∈ (Ring βˆ– NzRing)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍))
169, 11, 15syl2anc 585 . . . . 5 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) ∈ (π‘Ÿ RingHom 𝑍))
17 simpr 486 . . . . . 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 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ π‘ˆ ∈ 𝑉)
19 eqid 2737 . . . . . . . . . 10 (Hom β€˜πΆ) = (Hom β€˜πΆ)
20 simpr 486 . . . . . . . . . 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 3927 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑍 ∈ Ring)
2321, 22elind 4159 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑍 ∈ (π‘ˆ ∩ Ring))
2423, 4eleqtrrd 2841 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ (Baseβ€˜πΆ))
2524adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ 𝑍 ∈ (Baseβ€˜πΆ))
261, 2, 18, 19, 20, 25ringchom 46385 . . . . . . . . 9 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘Ÿ(Hom β€˜πΆ)𝑍) = (π‘Ÿ RingHom 𝑍))
2726eqcomd 2743 . . . . . . . 8 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (π‘Ÿ RingHom 𝑍) = (π‘Ÿ(Hom β€˜πΆ)𝑍))
2827eleq2d 2824 . . . . . . 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 478 . . . . . 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 2824 . . . . . . . . . 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 2737 . . . . . . . . . . 11 (Baseβ€˜π‘) = (Baseβ€˜π‘)
3212, 31rhmf 20167 . . . . . . . . . 10 (β„Ž ∈ (π‘Ÿ RingHom 𝑍) β†’ β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘))
3330, 32syl6bi 253 . . . . . . . . 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 482 . . . . . . . 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 6673 . . . . . . . . . . 11 (β„Ž:(Baseβ€˜π‘Ÿ)⟢(Baseβ€˜π‘) β†’ β„Ž Fn (Baseβ€˜π‘Ÿ))
3635adantl 483 . . . . . . . . . 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 6860 . . . . . . . . . . . 12 (0gβ€˜π‘) ∈ V
3837, 14fnmpti 6649 . . . . . . . . . . 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 46243 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring βˆ– NzRing) β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4241adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜πΆ)) β†’ (Baseβ€˜π‘) = {(0gβ€˜π‘)})
4342feq3d 6660 . . . . . . . . . . . . . 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 7115 . . . . . . . . . . . . . . 15 ((β„Ž:(Baseβ€˜π‘Ÿ)⟢{(0gβ€˜π‘)} ∧ π‘Ž ∈ (Baseβ€˜π‘Ÿ)) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘))
4544ex 414 . . . . . . . . . . . . . 14 (β„Ž:(Baseβ€˜π‘Ÿ)⟢{(0gβ€˜π‘)} β†’ (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (β„Žβ€˜π‘Ž) = (0gβ€˜π‘)))
4643, 45syl6bi 253 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 2738 . . . . . . . . . . . . 13 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘)))
50 eqidd 2738 . . . . . . . . . . . . 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 6960 . . . . . . . . . . . 12 (π‘Ž ∈ (Baseβ€˜π‘Ÿ) β†’ ((π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ (0gβ€˜π‘))β€˜π‘Ž) = (0gβ€˜π‘))
5453adantl 483 . . . . . . . . . . 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 2780 . . . . . . . . . 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 6990 . . . . . . . . 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 414 . . . . . . . 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 1931 . . . . . 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 1129 . . . . 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 686 . . . 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 2826 . . . . 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 3669 . . . 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 3144 . 2 (πœ‘ β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜πΆ)βˆƒ!β„Ž β„Ž ∈ (π‘Ÿ(Hom β€˜πΆ)𝑍))
661ringccat 46396 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
673, 66syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
682, 19, 67, 24istermo 17890 . 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 257 1 (πœ‘ β†’ 𝑍 ∈ (TermOβ€˜πΆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆƒ!weu 2567  βˆ€wral 3065  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914  {csn 4591   ↦ cmpt 5193   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Hom chom 17151  0gc0g 17328  Catccat 17551  TermOctermo 17875  Ringcrg 19971   RingHom crh 20152  NzRingcnzr 20743  RingCatcringc 46375
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-xnn0 12493  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-hash 14238  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-hom 17164  df-cco 17165  df-0g 17330  df-cat 17555  df-cid 17556  df-homf 17557  df-ssc 17700  df-resc 17701  df-subc 17702  df-termo 17878  df-estrc 18017  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-grp 18758  df-minusg 18759  df-ghm 19013  df-mgp 19904  df-ur 19921  df-ring 19973  df-rnghom 20155  df-nzr 20744  df-ringc 46377
This theorem is referenced by:  nzerooringczr  46444
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