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Theorem signstres 33885
Description: Restriction of a zero skipping sign to a subword. (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsv.p ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
signsv.w π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
signsv.t 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
signsv.v 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(β™―β€˜π‘“))if(((π‘‡β€˜π‘“)β€˜π‘—) β‰  ((π‘‡β€˜π‘“)β€˜(𝑗 βˆ’ 1)), 1, 0))
Assertion
Ref Expression
signstres ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) = (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))
Distinct variable groups:   π‘Ž,𝑏, ⨣   𝑓,𝑖,𝑛,𝐹   𝑓,π‘Š,𝑖,𝑛   𝑓,𝑁,𝑖,𝑛
Allowed substitution hints:   ⨣ (𝑓,𝑖,𝑗,𝑛)   𝑇(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝐹(𝑗,π‘Ž,𝑏)   𝑁(𝑗,π‘Ž,𝑏)   𝑉(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   π‘Š(𝑗,π‘Ž,𝑏)
Proof of Theorem signstres
Dummy variables 𝑔 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 signsv.p . . . . . . . 8 ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
2 signsv.w . . . . . . . 8 π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
3 signsv.t . . . . . . . 8 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
4 signsv.v . . . . . . . 8 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(β™―β€˜π‘“))if(((π‘‡β€˜π‘“)β€˜π‘—) β‰  ((π‘‡β€˜π‘“)β€˜(𝑗 βˆ’ 1)), 1, 0))
51, 2, 3, 4signstf 33876 . . . . . . 7 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜πΉ) ∈ Word ℝ)
6 wrdf 14474 . . . . . . 7 ((π‘‡β€˜πΉ) ∈ Word ℝ β†’ (π‘‡β€˜πΉ):(0..^(β™―β€˜(π‘‡β€˜πΉ)))βŸΆβ„)
7 ffn 6717 . . . . . . 7 ((π‘‡β€˜πΉ):(0..^(β™―β€˜(π‘‡β€˜πΉ)))βŸΆβ„ β†’ (π‘‡β€˜πΉ) Fn (0..^(β™―β€˜(π‘‡β€˜πΉ))))
85, 6, 73syl 18 . . . . . 6 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜πΉ) Fn (0..^(β™―β€˜(π‘‡β€˜πΉ))))
91, 2, 3, 4signstlen 33877 . . . . . . . 8 (𝐹 ∈ Word ℝ β†’ (β™―β€˜(π‘‡β€˜πΉ)) = (β™―β€˜πΉ))
109oveq2d 7428 . . . . . . 7 (𝐹 ∈ Word ℝ β†’ (0..^(β™―β€˜(π‘‡β€˜πΉ))) = (0..^(β™―β€˜πΉ)))
1110fneq2d 6643 . . . . . 6 (𝐹 ∈ Word ℝ β†’ ((π‘‡β€˜πΉ) Fn (0..^(β™―β€˜(π‘‡β€˜πΉ))) ↔ (π‘‡β€˜πΉ) Fn (0..^(β™―β€˜πΉ))))
128, 11mpbid 231 . . . . 5 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜πΉ) Fn (0..^(β™―β€˜πΉ)))
13 fnresin 32118 . . . . 5 ((π‘‡β€˜πΉ) Fn (0..^(β™―β€˜πΉ)) β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)))
1412, 13syl 17 . . . 4 (𝐹 ∈ Word ℝ β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)))
1514adantr 480 . . 3 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)))
16 elfzuz3 13503 . . . . . 6 (𝑁 ∈ (0...(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜π‘))
17 fzoss2 13665 . . . . . 6 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜π‘) β†’ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)))
1816, 17syl 17 . . . . 5 (𝑁 ∈ (0...(β™―β€˜πΉ)) β†’ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)))
1918adantl 481 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)))
20 incom 4201 . . . . . 6 ((0..^𝑁) ∩ (0..^(β™―β€˜πΉ))) = ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁))
21 df-ss 3965 . . . . . . 7 ((0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)) ↔ ((0..^𝑁) ∩ (0..^(β™―β€˜πΉ))) = (0..^𝑁))
2221biimpi 215 . . . . . 6 ((0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)) β†’ ((0..^𝑁) ∩ (0..^(β™―β€˜πΉ))) = (0..^𝑁))
2320, 22eqtr3id 2785 . . . . 5 ((0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)) β†’ ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)) = (0..^𝑁))
2423fneq2d 6643 . . . 4 ((0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)) ↔ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn (0..^𝑁)))
2519, 24syl 17 . . 3 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn ((0..^(β™―β€˜πΉ)) ∩ (0..^𝑁)) ↔ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn (0..^𝑁)))
2615, 25mpbid 231 . 2 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) Fn (0..^𝑁))
27 wrdres 32371 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 β†Ύ (0..^𝑁)) ∈ Word ℝ)
281, 2, 3, 4signstf 33876 . . . 4 ((𝐹 β†Ύ (0..^𝑁)) ∈ Word ℝ β†’ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) ∈ Word ℝ)
29 wrdf 14474 . . . 4 ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) ∈ Word ℝ β†’ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))):(0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))))βŸΆβ„)
30 ffn 6717 . . . 4 ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))):(0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))))βŸΆβ„ β†’ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) Fn (0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))))
3127, 28, 29, 304syl 19 . . 3 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) Fn (0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))))
321, 2, 3, 4signstlen 33877 . . . . . . 7 ((𝐹 β†Ύ (0..^𝑁)) ∈ Word ℝ β†’ (β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))) = (β™―β€˜(𝐹 β†Ύ (0..^𝑁))))
3327, 32syl 17 . . . . . 6 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))) = (β™―β€˜(𝐹 β†Ύ (0..^𝑁))))
34 wrdfn 14483 . . . . . . . 8 (𝐹 ∈ Word ℝ β†’ 𝐹 Fn (0..^(β™―β€˜πΉ)))
35 fnssres 6673 . . . . . . . 8 ((𝐹 Fn (0..^(β™―β€˜πΉ)) ∧ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ))) β†’ (𝐹 β†Ύ (0..^𝑁)) Fn (0..^𝑁))
3634, 18, 35syl2an 595 . . . . . . 7 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 β†Ύ (0..^𝑁)) Fn (0..^𝑁))
37 hashfn 14340 . . . . . . 7 ((𝐹 β†Ύ (0..^𝑁)) Fn (0..^𝑁) β†’ (β™―β€˜(𝐹 β†Ύ (0..^𝑁))) = (β™―β€˜(0..^𝑁)))
3836, 37syl 17 . . . . . 6 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 β†Ύ (0..^𝑁))) = (β™―β€˜(0..^𝑁)))
39 elfznn0 13599 . . . . . . . 8 (𝑁 ∈ (0...(β™―β€˜πΉ)) β†’ 𝑁 ∈ β„•0)
40 hashfzo0 14395 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (β™―β€˜(0..^𝑁)) = 𝑁)
4139, 40syl 17 . . . . . . 7 (𝑁 ∈ (0...(β™―β€˜πΉ)) β†’ (β™―β€˜(0..^𝑁)) = 𝑁)
4241adantl 481 . . . . . 6 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(0..^𝑁)) = 𝑁)
4333, 38, 423eqtrd 2775 . . . . 5 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))) = 𝑁)
4443oveq2d 7428 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))) = (0..^𝑁))
4544fneq2d 6643 . . 3 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) Fn (0..^(β™―β€˜(π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))) ↔ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) Fn (0..^𝑁)))
4631, 45mpbid 231 . 2 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))) Fn (0..^𝑁))
47 fvres 6910 . . . . 5 (π‘š ∈ (0..^𝑁) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁))β€˜π‘š) = ((π‘‡β€˜πΉ)β€˜π‘š))
4847ad3antlr 728 . . . 4 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁))β€˜π‘š) = ((π‘‡β€˜πΉ)β€˜π‘š))
49 simpr 484 . . . . . 6 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))
5049fveq2d 6895 . . . . 5 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ (π‘‡β€˜πΉ) = (π‘‡β€˜((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)))
5150fveq1d 6893 . . . 4 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ ((π‘‡β€˜πΉ)β€˜π‘š) = ((π‘‡β€˜((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))β€˜π‘š))
5227ad3antrrr 727 . . . . 5 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ (𝐹 β†Ύ (0..^𝑁)) ∈ Word ℝ)
53 simplr 766 . . . . 5 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ 𝑔 ∈ Word ℝ)
5438, 42eqtrd 2771 . . . . . . . . 9 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 β†Ύ (0..^𝑁))) = 𝑁)
5554oveq2d 7428 . . . . . . . 8 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(𝐹 β†Ύ (0..^𝑁)))) = (0..^𝑁))
5655eleq2d 2818 . . . . . . 7 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘š ∈ (0..^(β™―β€˜(𝐹 β†Ύ (0..^𝑁)))) ↔ π‘š ∈ (0..^𝑁)))
5756biimpar 477 . . . . . 6 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) β†’ π‘š ∈ (0..^(β™―β€˜(𝐹 β†Ύ (0..^𝑁)))))
5857ad2antrr 723 . . . . 5 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ π‘š ∈ (0..^(β™―β€˜(𝐹 β†Ύ (0..^𝑁)))))
591, 2, 3, 4signstfvc 33884 . . . . 5 (((𝐹 β†Ύ (0..^𝑁)) ∈ Word ℝ ∧ 𝑔 ∈ Word ℝ ∧ π‘š ∈ (0..^(β™―β€˜(𝐹 β†Ύ (0..^𝑁))))) β†’ ((π‘‡β€˜((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))β€˜π‘š) = ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))β€˜π‘š))
6052, 53, 58, 59syl3anc 1370 . . . 4 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ ((π‘‡β€˜((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))β€˜π‘š) = ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))β€˜π‘š))
6148, 51, 603eqtrd 2775 . . 3 (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔)) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁))β€˜π‘š) = ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))β€˜π‘š))
62 wrdsplex 32372 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ βˆƒπ‘” ∈ Word ℝ𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))
6362adantr 480 . . 3 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) β†’ βˆƒπ‘” ∈ Word ℝ𝐹 = ((𝐹 β†Ύ (0..^𝑁)) ++ 𝑔))
6461, 63r19.29a 3161 . 2 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) ∧ π‘š ∈ (0..^𝑁)) β†’ (((π‘‡β€˜πΉ) β†Ύ (0..^𝑁))β€˜π‘š) = ((π‘‡β€˜(𝐹 β†Ύ (0..^𝑁)))β€˜π‘š))
6526, 46, 64eqfnfvd 7035 1 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(β™―β€˜πΉ))) β†’ ((π‘‡β€˜πΉ) β†Ύ (0..^𝑁)) = (π‘‡β€˜(𝐹 β†Ύ (0..^𝑁))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   ∩ cin 3947   βŠ† wss 3948  ifcif 4528  {cpr 4630  {ctp 4632  βŸ¨cop 4634   ↦ cmpt 5231   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  β„cr 11113  0cc0 11114  1c1 11115   βˆ’ cmin 11449  -cneg 11450  β„•0cn0 12477  β„€β‰₯cuz 12827  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Word cword 14469   ++ cconcat 14525  sgncsgn 15038  Ξ£csu 15637  ndxcnx 17131  Basecbs 17149  +gcplusg 17202   Ξ£g cgsu 17391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-seq 13972  df-hash 14296  df-word 14470  df-lsw 14518  df-concat 14526  df-s1 14551  df-substr 14596  df-pfx 14626  df-sgn 15039  df-struct 17085  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-0g 17392  df-gsum 17393  df-mgm 18566  df-sgrp 18645  df-mnd 18661
This theorem is referenced by: (None)
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