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Theorem cutpw2 28432
Description: A cut expression for inverses of powers of two. (Contributed by Scott Fenton, 7-Aug-2025.)
Assertion
Ref Expression
cutpw2 (𝑁 ∈ ℕ0s → ( 1s /su (2ss(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑁))}))

Proof of Theorem cutpw2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . . . . 7 (𝑚 = 0s → (𝑚 +s 1s ) = ( 0s +s 1s ))
2 1sno 27887 . . . . . . . 8 1s No
3 addslid 28016 . . . . . . . 8 ( 1s No → ( 0s +s 1s ) = 1s )
42, 3ax-mp 5 . . . . . . 7 ( 0s +s 1s ) = 1s
51, 4eqtrdi 2791 . . . . . 6 (𝑚 = 0s → (𝑚 +s 1s ) = 1s )
65oveq2d 7447 . . . . 5 (𝑚 = 0s → (2ss(𝑚 +s 1s )) = (2ss 1s ))
7 2sno 28418 . . . . . 6 2s No
8 exps1 28426 . . . . . 6 (2s No → (2ss 1s ) = 2s)
97, 8ax-mp 5 . . . . 5 (2ss 1s ) = 2s
106, 9eqtrdi 2791 . . . 4 (𝑚 = 0s → (2ss(𝑚 +s 1s )) = 2s)
1110oveq2d 7447 . . 3 (𝑚 = 0s → ( 1s /su (2ss(𝑚 +s 1s ))) = ( 1s /su 2s))
12 oveq2 7439 . . . . . . . 8 (𝑚 = 0s → (2ss𝑚) = (2ss 0s ))
13 exps0 28425 . . . . . . . . 9 (2s No → (2ss 0s ) = 1s )
147, 13ax-mp 5 . . . . . . . 8 (2ss 0s ) = 1s
1512, 14eqtrdi 2791 . . . . . . 7 (𝑚 = 0s → (2ss𝑚) = 1s )
1615oveq2d 7447 . . . . . 6 (𝑚 = 0s → ( 1s /su (2ss𝑚)) = ( 1s /su 1s ))
17 divs1 28244 . . . . . . 7 ( 1s No → ( 1s /su 1s ) = 1s )
182, 17ax-mp 5 . . . . . 6 ( 1s /su 1s ) = 1s
1916, 18eqtrdi 2791 . . . . 5 (𝑚 = 0s → ( 1s /su (2ss𝑚)) = 1s )
2019sneqd 4643 . . . 4 (𝑚 = 0s → {( 1s /su (2ss𝑚))} = { 1s })
2120oveq2d 7447 . . 3 (𝑚 = 0s → ({ 0s } |s {( 1s /su (2ss𝑚))}) = ({ 0s } |s { 1s }))
2211, 21eqeq12d 2751 . 2 (𝑚 = 0s → (( 1s /su (2ss(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑚))}) ↔ ( 1s /su 2s) = ({ 0s } |s { 1s })))
23 oveq1 7438 . . . . 5 (𝑚 = 𝑛 → (𝑚 +s 1s ) = (𝑛 +s 1s ))
2423oveq2d 7447 . . . 4 (𝑚 = 𝑛 → (2ss(𝑚 +s 1s )) = (2ss(𝑛 +s 1s )))
2524oveq2d 7447 . . 3 (𝑚 = 𝑛 → ( 1s /su (2ss(𝑚 +s 1s ))) = ( 1s /su (2ss(𝑛 +s 1s ))))
26 oveq2 7439 . . . . . 6 (𝑚 = 𝑛 → (2ss𝑚) = (2ss𝑛))
2726oveq2d 7447 . . . . 5 (𝑚 = 𝑛 → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss𝑛)))
2827sneqd 4643 . . . 4 (𝑚 = 𝑛 → {( 1s /su (2ss𝑚))} = {( 1s /su (2ss𝑛))})
2928oveq2d 7447 . . 3 (𝑚 = 𝑛 → ({ 0s } |s {( 1s /su (2ss𝑚))}) = ({ 0s } |s {( 1s /su (2ss𝑛))}))
3025, 29eqeq12d 2751 . 2 (𝑚 = 𝑛 → (( 1s /su (2ss(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑚))}) ↔ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})))
31 oveq1 7438 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (𝑚 +s 1s ) = ((𝑛 +s 1s ) +s 1s ))
3231oveq2d 7447 . . . 4 (𝑚 = (𝑛 +s 1s ) → (2ss(𝑚 +s 1s )) = (2ss((𝑛 +s 1s ) +s 1s )))
3332oveq2d 7447 . . 3 (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2ss(𝑚 +s 1s ))) = ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))))
34 oveq2 7439 . . . . . 6 (𝑚 = (𝑛 +s 1s ) → (2ss𝑚) = (2ss(𝑛 +s 1s )))
3534oveq2d 7447 . . . . 5 (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss(𝑛 +s 1s ))))
3635sneqd 4643 . . . 4 (𝑚 = (𝑛 +s 1s ) → {( 1s /su (2ss𝑚))} = {( 1s /su (2ss(𝑛 +s 1s )))})
3736oveq2d 7447 . . 3 (𝑚 = (𝑛 +s 1s ) → ({ 0s } |s {( 1s /su (2ss𝑚))}) = ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))}))
3833, 37eqeq12d 2751 . 2 (𝑚 = (𝑛 +s 1s ) → (( 1s /su (2ss(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑚))}) ↔ ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))})))
39 oveq1 7438 . . . . 5 (𝑚 = 𝑁 → (𝑚 +s 1s ) = (𝑁 +s 1s ))
4039oveq2d 7447 . . . 4 (𝑚 = 𝑁 → (2ss(𝑚 +s 1s )) = (2ss(𝑁 +s 1s )))
4140oveq2d 7447 . . 3 (𝑚 = 𝑁 → ( 1s /su (2ss(𝑚 +s 1s ))) = ( 1s /su (2ss(𝑁 +s 1s ))))
42 oveq2 7439 . . . . . 6 (𝑚 = 𝑁 → (2ss𝑚) = (2ss𝑁))
4342oveq2d 7447 . . . . 5 (𝑚 = 𝑁 → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss𝑁)))
4443sneqd 4643 . . . 4 (𝑚 = 𝑁 → {( 1s /su (2ss𝑚))} = {( 1s /su (2ss𝑁))})
4544oveq2d 7447 . . 3 (𝑚 = 𝑁 → ({ 0s } |s {( 1s /su (2ss𝑚))}) = ({ 0s } |s {( 1s /su (2ss𝑁))}))
4641, 45eqeq12d 2751 . 2 (𝑚 = 𝑁 → (( 1s /su (2ss(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑚))}) ↔ ( 1s /su (2ss(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑁))})))
47 nohalf 28422 . 2 ( 1s /su 2s) = ({ 0s } |s { 1s })
48 peano2n0s 28350 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
49 expsp1 28427 . . . . . . . . . 10 ((2s No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2ss((𝑛 +s 1s ) +s 1s )) = ((2ss(𝑛 +s 1s )) ·s 2s))
507, 49mpan 690 . . . . . . . . 9 ((𝑛 +s 1s ) ∈ ℕ0s → (2ss((𝑛 +s 1s ) +s 1s )) = ((2ss(𝑛 +s 1s )) ·s 2s))
5148, 50syl 17 . . . . . . . 8 (𝑛 ∈ ℕ0s → (2ss((𝑛 +s 1s ) +s 1s )) = ((2ss(𝑛 +s 1s )) ·s 2s))
5251oveq2d 7447 . . . . . . 7 (𝑛 ∈ ℕ0s → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = ( 1s /su ((2ss(𝑛 +s 1s )) ·s 2s)))
532a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ0s → 1s No )
54 expscl 28428 . . . . . . . . . 10 ((2s No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2ss(𝑛 +s 1s )) ∈ No )
557, 54mpan 690 . . . . . . . . 9 ((𝑛 +s 1s ) ∈ ℕ0s → (2ss(𝑛 +s 1s )) ∈ No )
5648, 55syl 17 . . . . . . . 8 (𝑛 ∈ ℕ0s → (2ss(𝑛 +s 1s )) ∈ No )
577a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ0s → 2s No )
58 2ne0s 28419 . . . . . . . . . 10 2s ≠ 0s
59 expsne0 28429 . . . . . . . . . 10 ((2s No ∧ 2s ≠ 0s ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2ss(𝑛 +s 1s )) ≠ 0s )
607, 58, 59mp3an12 1450 . . . . . . . . 9 ((𝑛 +s 1s ) ∈ ℕ0s → (2ss(𝑛 +s 1s )) ≠ 0s )
6148, 60syl 17 . . . . . . . 8 (𝑛 ∈ ℕ0s → (2ss(𝑛 +s 1s )) ≠ 0s )
6258a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ0s → 2s ≠ 0s )
6353, 56, 57, 61, 62divdivs1d 28272 . . . . . . 7 (𝑛 ∈ ℕ0s → (( 1s /su (2ss(𝑛 +s 1s ))) /su 2s) = ( 1s /su ((2ss(𝑛 +s 1s )) ·s 2s)))
6452, 63eqtr4d 2778 . . . . . 6 (𝑛 ∈ ℕ0s → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = (( 1s /su (2ss(𝑛 +s 1s ))) /su 2s))
6553, 56, 61divscld 28263 . . . . . . . 8 (𝑛 ∈ ℕ0s → ( 1s /su (2ss(𝑛 +s 1s ))) ∈ No )
66 addslid 28016 . . . . . . . 8 (( 1s /su (2ss(𝑛 +s 1s ))) ∈ No → ( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) = ( 1s /su (2ss(𝑛 +s 1s ))))
6765, 66syl 17 . . . . . . 7 (𝑛 ∈ ℕ0s → ( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) = ( 1s /su (2ss(𝑛 +s 1s ))))
6867oveq1d 7446 . . . . . 6 (𝑛 ∈ ℕ0s → (( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) /su 2s) = (( 1s /su (2ss(𝑛 +s 1s ))) /su 2s))
6964, 68eqtr4d 2778 . . . . 5 (𝑛 ∈ ℕ0s → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = (( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) /su 2s))
7069adantr 480 . . . 4 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = (( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) /su 2s))
71 0sno 27886 . . . . . 6 0s No
7271a1i 11 . . . . 5 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → 0s No )
732a1i 11 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → 1s No )
7456adantr 480 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → (2ss(𝑛 +s 1s )) ∈ No )
7561adantr 480 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → (2ss(𝑛 +s 1s )) ≠ 0s )
7673, 74, 75divscld 28263 . . . . 5 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 1s /su (2ss(𝑛 +s 1s ))) ∈ No )
77 muls02 28182 . . . . . . . 8 ((2ss(𝑛 +s 1s )) ∈ No → ( 0s ·s (2ss(𝑛 +s 1s ))) = 0s )
7874, 77syl 17 . . . . . . 7 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 0s ·s (2ss(𝑛 +s 1s ))) = 0s )
79 0slt1s 27889 . . . . . . 7 0s <s 1s
8078, 79eqbrtrdi 5187 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 0s ·s (2ss(𝑛 +s 1s ))) <s 1s )
81 2nns 28417 . . . . . . . . . . 11 2s ∈ ℕs
82 nnsgt0 28357 . . . . . . . . . . 11 (2s ∈ ℕs → 0s <s 2s)
8381, 82ax-mp 5 . . . . . . . . . 10 0s <s 2s
84 expsgt0 28430 . . . . . . . . . 10 ((2s No ∧ (𝑛 +s 1s ) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2ss(𝑛 +s 1s )))
857, 83, 84mp3an13 1451 . . . . . . . . 9 ((𝑛 +s 1s ) ∈ ℕ0s → 0s <s (2ss(𝑛 +s 1s )))
8648, 85syl 17 . . . . . . . 8 (𝑛 ∈ ℕ0s → 0s <s (2ss(𝑛 +s 1s )))
8786adantr 480 . . . . . . 7 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → 0s <s (2ss(𝑛 +s 1s )))
8872, 73, 74, 87sltmuldivd 28268 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → (( 0s ·s (2ss(𝑛 +s 1s ))) <s 1s ↔ 0s <s ( 1s /su (2ss(𝑛 +s 1s )))))
8980, 88mpbid 232 . . . . 5 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → 0s <s ( 1s /su (2ss(𝑛 +s 1s ))))
90 muls01 28153 . . . . . . . . . . 11 (2s No → (2s ·s 0s ) = 0s )
917, 90ax-mp 5 . . . . . . . . . 10 (2s ·s 0s ) = 0s
9291sneqi 4642 . . . . . . . . 9 {(2s ·s 0s )} = { 0s }
9392a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ0s → {(2s ·s 0s )} = { 0s })
94 expsp1 28427 . . . . . . . . . . . . . 14 ((2s No 𝑛 ∈ ℕ0s) → (2ss(𝑛 +s 1s )) = ((2ss𝑛) ·s 2s))
957, 94mpan 690 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0s → (2ss(𝑛 +s 1s )) = ((2ss𝑛) ·s 2s))
9695oveq2d 7447 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0s → ( 1s /su (2ss(𝑛 +s 1s ))) = ( 1s /su ((2ss𝑛) ·s 2s)))
97 expscl 28428 . . . . . . . . . . . . . 14 ((2s No 𝑛 ∈ ℕ0s) → (2ss𝑛) ∈ No )
987, 97mpan 690 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0s → (2ss𝑛) ∈ No )
99 expsne0 28429 . . . . . . . . . . . . . 14 ((2s No ∧ 2s ≠ 0s𝑛 ∈ ℕ0s) → (2ss𝑛) ≠ 0s )
1007, 58, 99mp3an12 1450 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0s → (2ss𝑛) ≠ 0s )
10153, 98, 57, 100, 62divdivs1d 28272 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0s → (( 1s /su (2ss𝑛)) /su 2s) = ( 1s /su ((2ss𝑛) ·s 2s)))
10296, 101eqtr4d 2778 . . . . . . . . . . 11 (𝑛 ∈ ℕ0s → ( 1s /su (2ss(𝑛 +s 1s ))) = (( 1s /su (2ss𝑛)) /su 2s))
103102oveq2d 7447 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → (2s ·s ( 1s /su (2ss(𝑛 +s 1s )))) = (2s ·s (( 1s /su (2ss𝑛)) /su 2s)))
10453, 98, 100divscld 28263 . . . . . . . . . . 11 (𝑛 ∈ ℕ0s → ( 1s /su (2ss𝑛)) ∈ No )
105104, 57, 62divscan2d 28264 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → (2s ·s (( 1s /su (2ss𝑛)) /su 2s)) = ( 1s /su (2ss𝑛)))
106103, 105eqtrd 2775 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (2s ·s ( 1s /su (2ss(𝑛 +s 1s )))) = ( 1s /su (2ss𝑛)))
107106sneqd 4643 . . . . . . . 8 (𝑛 ∈ ℕ0s → {(2s ·s ( 1s /su (2ss(𝑛 +s 1s ))))} = {( 1s /su (2ss𝑛))})
10893, 107oveq12d 7449 . . . . . . 7 (𝑛 ∈ ℕ0s → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2ss(𝑛 +s 1s ))))}) = ({ 0s } |s {( 1s /su (2ss𝑛))}))
109 eqcom 2742 . . . . . . . 8 (( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))}) ↔ ({ 0s } |s {( 1s /su (2ss𝑛))}) = ( 1s /su (2ss(𝑛 +s 1s ))))
110109biimpi 216 . . . . . . 7 (( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))}) → ({ 0s } |s {( 1s /su (2ss𝑛))}) = ( 1s /su (2ss(𝑛 +s 1s ))))
111108, 110sylan9eq 2795 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2ss(𝑛 +s 1s ))))}) = ( 1s /su (2ss(𝑛 +s 1s ))))
11276, 66syl 17 . . . . . 6 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) = ( 1s /su (2ss(𝑛 +s 1s ))))
113111, 112eqtr4d 2778 . . . . 5 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2ss(𝑛 +s 1s ))))}) = ( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))))
114 eqid 2735 . . . . 5 ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))}) = ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))})
11572, 76, 89, 113, 114halfcut 28431 . . . 4 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))}) = (( 0s +s ( 1s /su (2ss(𝑛 +s 1s )))) /su 2s))
11670, 115eqtr4d 2778 . . 3 ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))})) → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))}))
117116ex 412 . 2 (𝑛 ∈ ℕ0s → (( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))}) → ( 1s /su (2ss((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2ss(𝑛 +s 1s )))})))
11822, 30, 38, 46, 47, 117n0sind 28352 1 (𝑁 ∈ ℕ0s → ( 1s /su (2ss(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑁))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  {csn 4631   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   |s cscut 27842   0s c0s 27882   1s c1s 27883   +s cadds 28007   ·s cmuls 28147   /su cdivs 28228  0scnn0s 28333  scnns 28334  2sc2s 28409  scexps 28411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-seqs 28305  df-n0s 28335  df-nns 28336  df-zs 28380  df-2s 28410  df-exps 28412
This theorem is referenced by:  pw2bday  28433
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