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Theorem zs12bday 28433
Description: A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.)
Assertion
Ref Expression
zs12bday (𝐴 ∈ ℤs[1/2] → ( bday 𝐴) ∈ ω)

Proof of Theorem zs12bday
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑡 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elzs12 28430 . 2 (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)))
2 fvoveq1 7468 . . . . . 6 (𝑧 = 𝑥 → ( bday ‘(𝑧 /su (2ss𝑦))) = ( bday ‘(𝑥 /su (2ss𝑦))))
32eleq1d 2823 . . . . 5 (𝑧 = 𝑥 → (( bday ‘(𝑧 /su (2ss𝑦))) ∈ ω ↔ ( bday ‘(𝑥 /su (2ss𝑦))) ∈ ω))
4 oveq2 7453 . . . . . . . . . . . 12 (𝑚 = 0s → (2ss𝑚) = (2ss 0s ))
5 2sno 28412 . . . . . . . . . . . . 13 2s No
6 exps0 28419 . . . . . . . . . . . . 13 (2s No → (2ss 0s ) = 1s )
75, 6ax-mp 5 . . . . . . . . . . . 12 (2ss 0s ) = 1s
84, 7eqtrdi 2790 . . . . . . . . . . 11 (𝑚 = 0s → (2ss𝑚) = 1s )
98oveq2d 7461 . . . . . . . . . 10 (𝑚 = 0s → (𝑧 /su (2ss𝑚)) = (𝑧 /su 1s ))
109fveq2d 6923 . . . . . . . . 9 (𝑚 = 0s → ( bday ‘(𝑧 /su (2ss𝑚))) = ( bday ‘(𝑧 /su 1s )))
1110eleq1d 2823 . . . . . . . 8 (𝑚 = 0s → (( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ( bday ‘(𝑧 /su 1s )) ∈ ω))
1211ralbidv 3180 . . . . . . 7 (𝑚 = 0s → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su 1s )) ∈ ω))
13 oveq2 7453 . . . . . . . . . . 11 (𝑚 = 𝑛 → (2ss𝑚) = (2ss𝑛))
1413oveq2d 7461 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑧 /su (2ss𝑚)) = (𝑧 /su (2ss𝑛)))
1514fveq2d 6923 . . . . . . . . 9 (𝑚 = 𝑛 → ( bday ‘(𝑧 /su (2ss𝑚))) = ( bday ‘(𝑧 /su (2ss𝑛))))
1615eleq1d 2823 . . . . . . . 8 (𝑚 = 𝑛 → (( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω))
1716ralbidv 3180 . . . . . . 7 (𝑚 = 𝑛 → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω))
18 oveq2 7453 . . . . . . . . . . . 12 (𝑚 = (𝑛 +s 1s ) → (2ss𝑚) = (2ss(𝑛 +s 1s )))
1918oveq2d 7461 . . . . . . . . . . 11 (𝑚 = (𝑛 +s 1s ) → (𝑧 /su (2ss𝑚)) = (𝑧 /su (2ss(𝑛 +s 1s ))))
2019fveq2d 6923 . . . . . . . . . 10 (𝑚 = (𝑛 +s 1s ) → ( bday ‘(𝑧 /su (2ss𝑚))) = ( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))))
2120eleq1d 2823 . . . . . . . . 9 (𝑚 = (𝑛 +s 1s ) → (( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))) ∈ ω))
2221ralbidv 3180 . . . . . . . 8 (𝑚 = (𝑛 +s 1s ) → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))) ∈ ω))
23 fvoveq1 7468 . . . . . . . . . 10 (𝑧 = 𝑤 → ( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))) = ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))))
2423eleq1d 2823 . . . . . . . . 9 (𝑧 = 𝑤 → (( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))) ∈ ω ↔ ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
2524cbvralvw 3238 . . . . . . . 8 (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss(𝑛 +s 1s )))) ∈ ω ↔ ∀𝑤 ∈ ℤs ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω)
2622, 25bitrdi 287 . . . . . . 7 (𝑚 = (𝑛 +s 1s ) → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ∀𝑤 ∈ ℤs ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
27 oveq2 7453 . . . . . . . . . . 11 (𝑚 = 𝑦 → (2ss𝑚) = (2ss𝑦))
2827oveq2d 7461 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑧 /su (2ss𝑚)) = (𝑧 /su (2ss𝑦)))
2928fveq2d 6923 . . . . . . . . 9 (𝑚 = 𝑦 → ( bday ‘(𝑧 /su (2ss𝑚))) = ( bday ‘(𝑧 /su (2ss𝑦))))
3029eleq1d 2823 . . . . . . . 8 (𝑚 = 𝑦 → (( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ( bday ‘(𝑧 /su (2ss𝑦))) ∈ ω))
3130ralbidv 3180 . . . . . . 7 (𝑚 = 𝑦 → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑚))) ∈ ω ↔ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑦))) ∈ ω))
32 zno 28377 . . . . . . . . . . 11 (𝑧 ∈ ℤs𝑧 No )
33 divs1 28238 . . . . . . . . . . 11 (𝑧 No → (𝑧 /su 1s ) = 𝑧)
3432, 33syl 17 . . . . . . . . . 10 (𝑧 ∈ ℤs → (𝑧 /su 1s ) = 𝑧)
3534fveq2d 6923 . . . . . . . . 9 (𝑧 ∈ ℤs → ( bday ‘(𝑧 /su 1s )) = ( bday 𝑧))
36 zsbday 28401 . . . . . . . . 9 (𝑧 ∈ ℤs → ( bday 𝑧) ∈ ω)
3735, 36eqeltrd 2838 . . . . . . . 8 (𝑧 ∈ ℤs → ( bday ‘(𝑧 /su 1s )) ∈ ω)
3837rgen 3065 . . . . . . 7 𝑧 ∈ ℤs ( bday ‘(𝑧 /su 1s )) ∈ ω
39 zseo 28415 . . . . . . . . . 10 (𝑤 ∈ ℤs → (∃𝑡 ∈ ℤs 𝑤 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℤs 𝑤 = ((2s ·s 𝑡) +s 1s )))
40 expsp1 28421 . . . . . . . . . . . . . . . . . . . . 21 ((2s No 𝑛 ∈ ℕ0s) → (2ss(𝑛 +s 1s )) = ((2ss𝑛) ·s 2s))
415, 40mpan 689 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0s → (2ss(𝑛 +s 1s )) = ((2ss𝑛) ·s 2s))
4241adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (2ss(𝑛 +s 1s )) = ((2ss𝑛) ·s 2s))
4342oveq2d 7461 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ((2s ·s 𝑡) /su (2ss(𝑛 +s 1s ))) = ((2s ·s 𝑡) /su ((2ss𝑛) ·s 2s)))
445a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → 2s No )
45 zno 28377 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ℤs𝑡 No )
4645adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → 𝑡 No )
4744, 46mulscld 28170 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (2s ·s 𝑡) ∈ No )
48 expscl 28422 . . . . . . . . . . . . . . . . . . . . 21 ((2s No 𝑛 ∈ ℕ0s) → (2ss𝑛) ∈ No )
495, 48mpan 689 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0s → (2ss𝑛) ∈ No )
5049adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (2ss𝑛) ∈ No )
51 2ne0s 28413 . . . . . . . . . . . . . . . . . . . . 21 2s ≠ 0s
52 expsne0 28423 . . . . . . . . . . . . . . . . . . . . 21 ((2s No ∧ 2s ≠ 0s𝑛 ∈ ℕ0s) → (2ss𝑛) ≠ 0s )
535, 51, 52mp3an12 1451 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0s → (2ss𝑛) ≠ 0s )
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (2ss𝑛) ≠ 0s )
5551a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → 2s ≠ 0s )
5647, 50, 44, 54, 55divdivs1d 28266 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (((2s ·s 𝑡) /su (2ss𝑛)) /su 2s) = ((2s ·s 𝑡) /su ((2ss𝑛) ·s 2s)))
5744, 46, 50, 54divsassd 28264 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ((2s ·s 𝑡) /su (2ss𝑛)) = (2s ·s (𝑡 /su (2ss𝑛))))
5857oveq1d 7460 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (((2s ·s 𝑡) /su (2ss𝑛)) /su 2s) = ((2s ·s (𝑡 /su (2ss𝑛))) /su 2s))
5943, 56, 583eqtr2d 2780 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ((2s ·s 𝑡) /su (2ss(𝑛 +s 1s ))) = ((2s ·s (𝑡 /su (2ss𝑛))) /su 2s))
6046, 50, 54divscld 28257 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → (𝑡 /su (2ss𝑛)) ∈ No )
6160, 44, 55divscan3d 28269 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ((2s ·s (𝑡 /su (2ss𝑛))) /su 2s) = (𝑡 /su (2ss𝑛)))
6259, 61eqtrd 2774 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ((2s ·s 𝑡) /su (2ss(𝑛 +s 1s ))) = (𝑡 /su (2ss𝑛)))
6362fveq2d 6923 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0s𝑡 ∈ ℤs) → ( bday ‘((2s ·s 𝑡) /su (2ss(𝑛 +s 1s )))) = ( bday ‘(𝑡 /su (2ss𝑛))))
6463adantlr 714 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘((2s ·s 𝑡) /su (2ss(𝑛 +s 1s )))) = ( bday ‘(𝑡 /su (2ss𝑛))))
65 fvoveq1 7468 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑡 → ( bday ‘(𝑧 /su (2ss𝑛))) = ( bday ‘(𝑡 /su (2ss𝑛))))
6665eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑡 → (( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω ↔ ( bday ‘(𝑡 /su (2ss𝑛))) ∈ ω))
6766rspccva 3630 . . . . . . . . . . . . . . 15 ((∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω ∧ 𝑡 ∈ ℤs) → ( bday ‘(𝑡 /su (2ss𝑛))) ∈ ω)
6867adantll 713 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘(𝑡 /su (2ss𝑛))) ∈ ω)
6964, 68eqeltrd 2838 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘((2s ·s 𝑡) /su (2ss(𝑛 +s 1s )))) ∈ ω)
70 fvoveq1 7468 . . . . . . . . . . . . . 14 (𝑤 = (2s ·s 𝑡) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) = ( bday ‘((2s ·s 𝑡) /su (2ss(𝑛 +s 1s )))))
7170eleq1d 2823 . . . . . . . . . . . . 13 (𝑤 = (2s ·s 𝑡) → (( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω ↔ ( bday ‘((2s ·s 𝑡) /su (2ss(𝑛 +s 1s )))) ∈ ω))
7269, 71syl5ibrcom 247 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑤 = (2s ·s 𝑡) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
7372rexlimdva 3157 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) → (∃𝑡 ∈ ℤs 𝑤 = (2s ·s 𝑡) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
7445adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 𝑡 No )
75 no2times 28410 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 No → (2s ·s 𝑡) = (𝑡 +s 𝑡))
7674, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s 𝑡) = (𝑡 +s 𝑡))
7776oveq1d 7460 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2s ·s 𝑡) +s 1s ) = ((𝑡 +s 𝑡) +s 1s ))
78 1sno 27881 . . . . . . . . . . . . . . . . . . . . 21 1s No
7978a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 1s No )
8074, 74, 79addsassd 28048 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((𝑡 +s 𝑡) +s 1s ) = (𝑡 +s (𝑡 +s 1s )))
8177, 80eqtrd 2774 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2s ·s 𝑡) +s 1s ) = (𝑡 +s (𝑡 +s 1s )))
8281oveq1d 7460 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s ))) = ((𝑡 +s (𝑡 +s 1s )) /su (2ss(𝑛 +s 1s ))))
8374, 79addscld 28022 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑡 +s 1s ) ∈ No )
84 simpll 766 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 𝑛 ∈ ℕ0s)
8574sltp1d 28057 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 𝑡 <s (𝑡 +s 1s ))
86 2nns 28411 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2s ∈ ℕs
87 nnzs 28381 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (2s ∈ ℕs → 2s ∈ ℤs)
8886, 87mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 2s ∈ ℤs)
89 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 𝑡 ∈ ℤs)
9088, 89zmulscld 28392 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s 𝑡) ∈ ℤs)
9190znod 28378 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s 𝑡) ∈ No )
92 pncans 28111 . . . . . . . . . . . . . . . . . . . . . . 23 (((2s ·s 𝑡) ∈ No ∧ 1s No ) → (((2s ·s 𝑡) +s 1s ) -s 1s ) = (2s ·s 𝑡))
9391, 78, 92sylancl 585 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (((2s ·s 𝑡) +s 1s ) -s 1s ) = (2s ·s 𝑡))
9493eqcomd 2740 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s 𝑡) = (((2s ·s 𝑡) +s 1s ) -s 1s ))
9594sneqd 4660 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {(2s ·s 𝑡)} = {(((2s ·s 𝑡) +s 1s ) -s 1s )})
96 mulsrid 28148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2s No → (2s ·s 1s ) = 2s)
975, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 (2s ·s 1s ) = 2s
98 1p1e2s 28409 . . . . . . . . . . . . . . . . . . . . . . . 24 ( 1s +s 1s ) = 2s
9997, 98eqtr4i 2765 . . . . . . . . . . . . . . . . . . . . . . 23 (2s ·s 1s ) = ( 1s +s 1s )
10099oveq2i 7456 . . . . . . . . . . . . . . . . . . . . . 22 ((2s ·s 𝑡) +s (2s ·s 1s )) = ((2s ·s 𝑡) +s ( 1s +s 1s ))
1015a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 2s No )
102101, 74, 79addsdid 28191 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s (𝑡 +s 1s )) = ((2s ·s 𝑡) +s (2s ·s 1s )))
10391, 79, 79addsassd 28048 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (((2s ·s 𝑡) +s 1s ) +s 1s ) = ((2s ·s 𝑡) +s ( 1s +s 1s )))
104100, 102, 1033eqtr4a 2800 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2s ·s (𝑡 +s 1s )) = (((2s ·s 𝑡) +s 1s ) +s 1s ))
105104sneqd 4660 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {(2s ·s (𝑡 +s 1s ))} = {(((2s ·s 𝑡) +s 1s ) +s 1s )})
10695, 105oveq12d 7463 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ({(2s ·s 𝑡)} |s {(2s ·s (𝑡 +s 1s ))}) = ({(((2s ·s 𝑡) +s 1s ) -s 1s )} |s {(((2s ·s 𝑡) +s 1s ) +s 1s )}))
107 1zs 28386 . . . . . . . . . . . . . . . . . . . . . 22 1s ∈ ℤs
108107a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 1s ∈ ℤs)
10990, 108zaddscld 28390 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2s ·s 𝑡) +s 1s ) ∈ ℤs)
110 zscut 28402 . . . . . . . . . . . . . . . . . . . 20 (((2s ·s 𝑡) +s 1s ) ∈ ℤs → ((2s ·s 𝑡) +s 1s ) = ({(((2s ·s 𝑡) +s 1s ) -s 1s )} |s {(((2s ·s 𝑡) +s 1s ) +s 1s )}))
111109, 110syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2s ·s 𝑡) +s 1s ) = ({(((2s ·s 𝑡) +s 1s ) -s 1s )} |s {(((2s ·s 𝑡) +s 1s ) +s 1s )}))
112106, 111, 813eqtr2d 2780 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ({(2s ·s 𝑡)} |s {(2s ·s (𝑡 +s 1s ))}) = (𝑡 +s (𝑡 +s 1s )))
11374, 83, 84, 85, 112pw2cut 28429 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))}) = ((𝑡 +s (𝑡 +s 1s )) /su (2ss(𝑛 +s 1s ))))
11482, 113eqtr4d 2777 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s ))) = ({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))}))
115114fveq2d 6923 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) = ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})))
11649ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2ss𝑛) ∈ No )
11753ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (2ss𝑛) ≠ 0s )
11874, 116, 117divscld 28257 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑡 /su (2ss𝑛)) ∈ No )
11983, 116, 117divscld 28257 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((𝑡 +s 1s ) /su (2ss𝑛)) ∈ No )
12074, 116, 117divscan2d 28258 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2ss𝑛) ·s (𝑡 /su (2ss𝑛))) = 𝑡)
121120, 85eqbrtrd 5191 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ((2ss𝑛) ·s (𝑡 /su (2ss𝑛))) <s (𝑡 +s 1s ))
122 nnsgt0 28351 . . . . . . . . . . . . . . . . . . . . . . 23 (2s ∈ ℕs → 0s <s 2s)
12386, 122ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 0s <s 2s
124 expsgt0 28424 . . . . . . . . . . . . . . . . . . . . . 22 ((2s No 𝑛 ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2ss𝑛))
1255, 123, 124mp3an13 1452 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ0s → 0s <s (2ss𝑛))
126125ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → 0s <s (2ss𝑛))
127118, 83, 116, 126sltmuldiv2d 28263 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (((2ss𝑛) ·s (𝑡 /su (2ss𝑛))) <s (𝑡 +s 1s ) ↔ (𝑡 /su (2ss𝑛)) <s ((𝑡 +s 1s ) /su (2ss𝑛))))
128121, 127mpbid 232 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑡 /su (2ss𝑛)) <s ((𝑡 +s 1s ) /su (2ss𝑛)))
129118, 119, 128ssltsn 27846 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {(𝑡 /su (2ss𝑛))} <<s {((𝑡 +s 1s ) /su (2ss𝑛))})
130 imaundi 6180 . . . . . . . . . . . . . . . . . . . . . 22 ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) = (( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
131130unieqi 4943 . . . . . . . . . . . . . . . . . . . . 21 ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) = (( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
132 uniun 4956 . . . . . . . . . . . . . . . . . . . . 21 (( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))})) = ( ( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
133131, 132eqtri 2762 . . . . . . . . . . . . . . . . . . . 20 ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) = ( ( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
134 bdayfn 27827 . . . . . . . . . . . . . . . . . . . . . . . . 25 bday Fn No
135 fnsnfv 6999 . . . . . . . . . . . . . . . . . . . . . . . . 25 (( bday Fn No ∧ (𝑡 /su (2ss𝑛)) ∈ No ) → {( bday ‘(𝑡 /su (2ss𝑛)))} = ( bday “ {(𝑡 /su (2ss𝑛))}))
136134, 118, 135sylancr 586 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {( bday ‘(𝑡 /su (2ss𝑛)))} = ( bday “ {(𝑡 /su (2ss𝑛))}))
137136unieqd 4944 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {( bday ‘(𝑡 /su (2ss𝑛)))} = ( bday “ {(𝑡 /su (2ss𝑛))}))
138 fvex 6932 . . . . . . . . . . . . . . . . . . . . . . . 24 ( bday ‘(𝑡 /su (2ss𝑛))) ∈ V
139138unisn 4952 . . . . . . . . . . . . . . . . . . . . . . 23 {( bday ‘(𝑡 /su (2ss𝑛)))} = ( bday ‘(𝑡 /su (2ss𝑛)))
140137, 139eqtr3di 2789 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ {(𝑡 /su (2ss𝑛))}) = ( bday ‘(𝑡 /su (2ss𝑛))))
141140, 68eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ {(𝑡 /su (2ss𝑛))}) ∈ ω)
142 fnsnfv 6999 . . . . . . . . . . . . . . . . . . . . . . . . 25 (( bday Fn No ∧ ((𝑡 +s 1s ) /su (2ss𝑛)) ∈ No ) → {( bday ‘((𝑡 +s 1s ) /su (2ss𝑛)))} = ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
143134, 119, 142sylancr 586 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {( bday ‘((𝑡 +s 1s ) /su (2ss𝑛)))} = ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
144143unieqd 4944 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → {( bday ‘((𝑡 +s 1s ) /su (2ss𝑛)))} = ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}))
145 fvex 6932 . . . . . . . . . . . . . . . . . . . . . . . 24 ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛))) ∈ V
146145unisn 4952 . . . . . . . . . . . . . . . . . . . . . . 23 {( bday ‘((𝑡 +s 1s ) /su (2ss𝑛)))} = ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛)))
147144, 146eqtr3di 2789 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}) = ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛))))
148 fvoveq1 7468 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (𝑡 +s 1s ) → ( bday ‘(𝑧 /su (2ss𝑛))) = ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛))))
149148eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝑡 +s 1s ) → (( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω ↔ ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛))) ∈ ω))
150 simplr 768 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω)
15189, 108zaddscld 28390 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑡 +s 1s ) ∈ ℤs)
152149, 150, 151rspcdva 3632 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘((𝑡 +s 1s ) /su (2ss𝑛))) ∈ ω)
153147, 152eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}) ∈ ω)
154 omun 7922 . . . . . . . . . . . . . . . . . . . . 21 (( ( bday “ {(𝑡 /su (2ss𝑛))}) ∈ ω ∧ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))}) ∈ ω) → ( ( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
155141, 153, 154syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( ( bday “ {(𝑡 /su (2ss𝑛))}) ∪ ( bday “ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
156133, 155eqeltrid 2842 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
157 peano2 7925 . . . . . . . . . . . . . . . . . . 19 ( ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω → suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
158156, 157syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
159 nnon 7905 . . . . . . . . . . . . . . . . . 18 (suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω → suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On)
160158, 159syl 17 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On)
161 imassrn 6099 . . . . . . . . . . . . . . . . . . 19 ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ ran bday
162 bdayrn 27829 . . . . . . . . . . . . . . . . . . 19 ran bday = On
163161, 162sseqtri 4039 . . . . . . . . . . . . . . . . . 18 ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ On
164 onsucuni 7860 . . . . . . . . . . . . . . . . . 18 (( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ On → ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
165163, 164mp1i 13 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
166 scutbdaybnd 27869 . . . . . . . . . . . . . . . . 17 (({(𝑡 /su (2ss𝑛))} <<s {((𝑡 +s 1s ) /su (2ss𝑛))} ∧ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On ∧ ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))}))) → ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
167129, 160, 165, 166syl3anc 1371 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
168 bdayelon 27830 . . . . . . . . . . . . . . . . 17 ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On
169 onsssuc 6484 . . . . . . . . . . . . . . . . 17 ((( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On ∧ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ On) → (( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ↔ ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))}))))
170168, 160, 169sylancr 586 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ⊆ suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ↔ ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))}))))
171167, 170mpbid 232 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘({(𝑡 /su (2ss𝑛))} |s {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
172115, 171eqeltrd 2838 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})))
173 peano2 7925 . . . . . . . . . . . . . . 15 (suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω → suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
174158, 173syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω)
175 elnn 7910 . . . . . . . . . . . . . 14 ((( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∧ suc suc ( bday “ ({(𝑡 /su (2ss𝑛))} ∪ {((𝑡 +s 1s ) /su (2ss𝑛))})) ∈ ω) → ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) ∈ ω)
176172, 174, 175syl2anc 583 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) ∈ ω)
177 fvoveq1 7468 . . . . . . . . . . . . . 14 (𝑤 = ((2s ·s 𝑡) +s 1s ) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) = ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))))
178177eleq1d 2823 . . . . . . . . . . . . 13 (𝑤 = ((2s ·s 𝑡) +s 1s ) → (( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω ↔ ( bday ‘(((2s ·s 𝑡) +s 1s ) /su (2ss(𝑛 +s 1s )))) ∈ ω))
179176, 178syl5ibrcom 247 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) ∧ 𝑡 ∈ ℤs) → (𝑤 = ((2s ·s 𝑡) +s 1s ) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
180179rexlimdva 3157 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) → (∃𝑡 ∈ ℤs 𝑤 = ((2s ·s 𝑡) +s 1s ) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
18173, 180jaod 858 . . . . . . . . . 10 ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) → ((∃𝑡 ∈ ℤs 𝑤 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℤs 𝑤 = ((2s ·s 𝑡) +s 1s )) → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
18239, 181syl5 34 . . . . . . . . 9 ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) → (𝑤 ∈ ℤs → ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
183182ralrimiv 3147 . . . . . . . 8 ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω) → ∀𝑤 ∈ ℤs ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω)
184183ex 412 . . . . . . 7 (𝑛 ∈ ℕ0s → (∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑛))) ∈ ω → ∀𝑤 ∈ ℤs ( bday ‘(𝑤 /su (2ss(𝑛 +s 1s )))) ∈ ω))
18512, 17, 26, 31, 38, 184n0sind 28346 . . . . . 6 (𝑦 ∈ ℕ0s → ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑦))) ∈ ω)
186185adantl 481 . . . . 5 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → ∀𝑧 ∈ ℤs ( bday ‘(𝑧 /su (2ss𝑦))) ∈ ω)
187 simpl 482 . . . . 5 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → 𝑥 ∈ ℤs)
1883, 186, 187rspcdva 3632 . . . 4 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → ( bday ‘(𝑥 /su (2ss𝑦))) ∈ ω)
189 fveq2 6919 . . . . 5 (𝐴 = (𝑥 /su (2ss𝑦)) → ( bday 𝐴) = ( bday ‘(𝑥 /su (2ss𝑦))))
190189eleq1d 2823 . . . 4 (𝐴 = (𝑥 /su (2ss𝑦)) → (( bday 𝐴) ∈ ω ↔ ( bday ‘(𝑥 /su (2ss𝑦))) ∈ ω))
191188, 190syl5ibrcom 247 . . 3 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2ss𝑦)) → ( bday 𝐴) ∈ ω))
192191rexlimivv 3203 . 2 (∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)) → ( bday 𝐴) ∈ ω)
1931, 192sylbi 217 1 (𝐴 ∈ ℤs[1/2] → ( bday 𝐴) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846   = wceq 1537  wcel 2103  wne 2942  wral 3063  wrex 3072  cun 3968  wss 3970  {csn 4648   cuni 4931   class class class wbr 5169  ran crn 5700  cima 5702  Oncon0 6394  suc csuc 6396   Fn wfn 6567  cfv 6572  (class class class)co 7445  ωcom 7899   No csur 27693   <s cslt 27694   bday cbday 27695   <<s csslt 27834   |s cscut 27836   0s c0s 27876   1s c1s 27877   +s cadds 28001   -s csubs 28061   ·s cmuls 28141   /su cdivs 28222  0scnn0s 28327  scnns 28328  sczs 28373  2sc2s 28403  scexps 28405  s[1/2]czs12 28407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-dc 10511
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-oadd 8522  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sle 27799  df-sslt 27835  df-scut 27837  df-0s 27878  df-1s 27879  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec 27980  df-norec2 27991  df-adds 28002  df-negs 28062  df-subs 28063  df-muls 28142  df-divs 28223  df-seqs 28299  df-n0s 28329  df-nns 28330  df-zs 28374  df-2s 28404  df-exps 28406  df-zs12 28408
This theorem is referenced by: (None)
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