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Theorem axcaucvglemval 7898
Description: Lemma for axcaucvg 7901. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
Hypotheses
Ref Expression
axcaucvg.n 𝑁 = ∩ {π‘₯ ∣ (1 ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 + 1) ∈ π‘₯)}
axcaucvg.f (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
axcaucvg.cau (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 βˆ€π‘˜ ∈ 𝑁 (𝑛 <ℝ π‘˜ β†’ ((πΉβ€˜π‘›) <ℝ ((πΉβ€˜π‘˜) + (β„©π‘Ÿ ∈ ℝ (𝑛 Β· π‘Ÿ) = 1)) ∧ (πΉβ€˜π‘˜) <ℝ ((πΉβ€˜π‘›) + (β„©π‘Ÿ ∈ ℝ (𝑛 Β· π‘Ÿ) = 1)))))
axcaucvg.g 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
Assertion
Ref Expression
axcaucvglemval ((πœ‘ ∧ 𝐽 ∈ N) β†’ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(πΊβ€˜π½), 0R⟩)
Distinct variable groups:   𝑗,𝐹,𝑧   𝑧,𝐺   𝑗,𝐽,𝑙,𝑒,𝑧   πœ‘,𝑗   𝑦,𝑙,𝑒   π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧,𝑒,π‘˜,𝑛,π‘Ÿ,𝑙)   𝐹(π‘₯,𝑦,𝑒,π‘˜,𝑛,π‘Ÿ,𝑙)   𝐺(π‘₯,𝑦,𝑒,𝑗,π‘˜,𝑛,π‘Ÿ,𝑙)   𝐽(π‘₯,𝑦,π‘˜,𝑛,π‘Ÿ)   𝑁(π‘₯,𝑦,𝑧,𝑒,𝑗,π‘˜,𝑛,π‘Ÿ,𝑙)
Proof of Theorem axcaucvglemval
StepHypRef Expression
1 axcaucvg.g . . . . 5 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
21a1i 9 . . . 4 ((πœ‘ ∧ 𝐽 ∈ N) β†’ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩)))
3 opeq1 3780 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 β†’ βŸ¨π‘—, 1o⟩ = ⟨𝐽, 1o⟩)
43eceq1d 6573 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 β†’ [βŸ¨π‘—, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
54breq2d 4017 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 β†’ (𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q ))
65abbidv 2295 . . . . . . . . . . . . 13 (𝑗 = 𝐽 β†’ {𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q })
74breq1d 4015 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 β†’ ([βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒 ↔ [⟨𝐽, 1o⟩] ~Q <Q 𝑒))
87abbidv 2295 . . . . . . . . . . . . 13 (𝑗 = 𝐽 β†’ {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒} = {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒})
96, 8opeq12d 3788 . . . . . . . . . . . 12 (𝑗 = 𝐽 β†’ ⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩)
109oveq1d 5892 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ (⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P))
1110opeq1d 3786 . . . . . . . . . 10 (𝑗 = 𝐽 β†’ ⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩)
1211eceq1d 6573 . . . . . . . . 9 (𝑗 = 𝐽 β†’ [⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R )
1312opeq1d 3786 . . . . . . . 8 (𝑗 = 𝐽 β†’ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
1413fveq2d 5521 . . . . . . 7 (𝑗 = 𝐽 β†’ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
1514eqeq1d 2186 . . . . . 6 (𝑗 = 𝐽 β†’ ((πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩ ↔ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
1615riotabidv 5835 . . . . 5 (𝑗 = 𝐽 β†’ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) = (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
1716adantl 277 . . . 4 (((πœ‘ ∧ 𝐽 ∈ N) ∧ 𝑗 = 𝐽) β†’ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [βŸ¨π‘—, 1o⟩] ~Q }, {𝑒 ∣ [βŸ¨π‘—, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) = (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
18 simpr 110 . . . 4 ((πœ‘ ∧ 𝐽 ∈ N) β†’ 𝐽 ∈ N)
19 axcaucvg.n . . . . 5 𝑁 = ∩ {π‘₯ ∣ (1 ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 + 1) ∈ π‘₯)}
20 axcaucvg.f . . . . 5 (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
2119, 20axcaucvglemcl 7896 . . . 4 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) ∈ R)
222, 17, 18, 21fvmptd 5599 . . 3 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (πΊβ€˜π½) = (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩))
2322eqcomd 2183 . 2 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) = (πΊβ€˜π½))
2422, 21eqeltrd 2254 . . 3 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (πΊβ€˜π½) ∈ R)
2520adantr 276 . . . . . 6 ((πœ‘ ∧ 𝐽 ∈ N) β†’ 𝐹:π‘βŸΆβ„)
26 pitonn 7849 . . . . . . . 8 (𝐽 ∈ N β†’ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {π‘₯ ∣ (1 ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 + 1) ∈ π‘₯)})
2726, 19eleqtrrdi 2271 . . . . . . 7 (𝐽 ∈ N β†’ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
2827adantl 277 . . . . . 6 ((πœ‘ ∧ 𝐽 ∈ N) β†’ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
2925, 28ffvelcdmd 5654 . . . . 5 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
30 elrealeu 7830 . . . . 5 ((πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ ↔ βˆƒ!𝑧 ∈ R βŸ¨π‘§, 0R⟩ = (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
3129, 30sylib 122 . . . 4 ((πœ‘ ∧ 𝐽 ∈ N) β†’ βˆƒ!𝑧 ∈ R βŸ¨π‘§, 0R⟩ = (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
32 eqcom 2179 . . . . 5 (βŸ¨π‘§, 0R⟩ = (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩)
3332reubii 2663 . . . 4 (βˆƒ!𝑧 ∈ R βŸ¨π‘§, 0R⟩ = (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ βˆƒ!𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩)
3431, 33sylib 122 . . 3 ((πœ‘ ∧ 𝐽 ∈ N) β†’ βˆƒ!𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩)
35 opeq1 3780 . . . . 5 (𝑧 = (πΊβ€˜π½) β†’ βŸ¨π‘§, 0R⟩ = ⟨(πΊβ€˜π½), 0R⟩)
3635eqeq2d 2189 . . . 4 (𝑧 = (πΊβ€˜π½) β†’ ((πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩ ↔ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(πΊβ€˜π½), 0R⟩))
3736riota2 5855 . . 3 (((πΊβ€˜π½) ∈ R ∧ βˆƒ!𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) β†’ ((πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(πΊβ€˜π½), 0R⟩ ↔ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) = (πΊβ€˜π½)))
3824, 34, 37syl2anc 411 . 2 ((πœ‘ ∧ 𝐽 ∈ N) β†’ ((πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(πΊβ€˜π½), 0R⟩ ↔ (℩𝑧 ∈ R (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = βŸ¨π‘§, 0R⟩) = (πΊβ€˜π½)))
3923, 38mpbird 167 1 ((πœ‘ ∧ 𝐽 ∈ N) β†’ (πΉβ€˜βŸ¨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑒 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑒}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(πΊβ€˜π½), 0R⟩)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  βˆ€wral 2455  βˆƒ!wreu 2457  βŸ¨cop 3597  βˆ© cint 3846   class class class wbr 4005   ↦ cmpt 4066  βŸΆwf 5214  β€˜cfv 5218  β„©crio 5832  (class class class)co 5877  1oc1o 6412  [cec 6535  Ncnpi 7273   ~Q ceq 7280   <Q cltq 7286  1Pc1p 7293   +P cpp 7294   ~R cer 7297  Rcnr 7298  0Rc0r 7299  β„cr 7812  1c1 7814   + caddc 7816   <ℝ cltrr 7817   Β· cmul 7818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-enr 7727  df-nr 7728  df-plr 7729  df-0r 7732  df-1r 7733  df-c 7819  df-1 7821  df-r 7823  df-add 7824
This theorem is referenced by:  axcaucvglemcau  7899  axcaucvglemres  7900
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