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Theorem sticksstones17 40967
Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
Hypotheses
Ref Expression
sticksstones17.1 (πœ‘ β†’ 𝑁 ∈ β„•0)
sticksstones17.2 (πœ‘ β†’ 𝐾 ∈ β„•0)
sticksstones17.3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)}
sticksstones17.4 𝐡 = {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}
sticksstones17.5 (πœ‘ β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
sticksstones17.6 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
Assertion
Ref Expression
sticksstones17 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
Distinct variable groups:   𝐴,𝑏   𝐡,𝑏,𝑖,𝑦   𝑔,𝐾,𝑖,𝑦   𝑔,𝑁   β„Ž,𝑁   𝑆,β„Ž,𝑖   𝑔,𝑍,𝑖,𝑦   𝑔,𝑏   β„Ž,𝑏   πœ‘,𝑏,𝑖,𝑦
Allowed substitution hints:   πœ‘(𝑔,β„Ž)   𝐴(𝑦,𝑔,β„Ž,𝑖)   𝐡(𝑔,β„Ž)   𝑆(𝑦,𝑔,𝑏)   𝐺(𝑦,𝑔,β„Ž,𝑖,𝑏)   𝐾(β„Ž,𝑏)   𝑁(𝑦,𝑖,𝑏)   𝑍(β„Ž,𝑏)
Proof of Theorem sticksstones17
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 sticksstones17.4 . . . . . . . . . . . . . . . 16 𝐡 = {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}
21eqimssi 4041 . . . . . . . . . . . . . . 15 𝐡 βŠ† {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}
32a1i 11 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 βŠ† {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)})
43sseld 3980 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}))
54imp 407 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)})
6 vex 3478 . . . . . . . . . . . . 13 𝑏 ∈ V
7 feq1 6695 . . . . . . . . . . . . . 14 (β„Ž = 𝑏 β†’ (β„Ž:π‘†βŸΆβ„•0 ↔ 𝑏:π‘†βŸΆβ„•0))
8 simpl 483 . . . . . . . . . . . . . . . . 17 ((β„Ž = 𝑏 ∧ 𝑖 ∈ 𝑆) β†’ β„Ž = 𝑏)
98fveq1d 6890 . . . . . . . . . . . . . . . 16 ((β„Ž = 𝑏 ∧ 𝑖 ∈ 𝑆) β†’ (β„Žβ€˜π‘–) = (π‘β€˜π‘–))
109sumeq2dv 15645 . . . . . . . . . . . . . . 15 (β„Ž = 𝑏 β†’ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–))
1110eqeq1d 2734 . . . . . . . . . . . . . 14 (β„Ž = 𝑏 β†’ (Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁))
127, 11anbi12d 631 . . . . . . . . . . . . 13 (β„Ž = 𝑏 β†’ ((β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁) ↔ (𝑏:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁)))
136, 12elab 3667 . . . . . . . . . . . 12 (𝑏 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)} ↔ (𝑏:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁))
145, 13sylib 217 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑏:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁))
1514simpld 495 . . . . . . . . . 10 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏:π‘†βŸΆβ„•0)
1615adantr 481 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑏:π‘†βŸΆβ„•0)
17163impa 1110 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑏:π‘†βŸΆβ„•0)
18 sticksstones17.5 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
19 f1of 6830 . . . . . . . . . . . . 13 (𝑍:(1...𝐾)–1-1-onto→𝑆 β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
2018, 19syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
2120adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
2221adantr 481 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
23223impa 1110 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
24 simp3 1138 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑦 ∈ (1...𝐾))
2523, 24ffvelcdmd 7084 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜π‘¦) ∈ 𝑆)
2617, 25ffvelcdmd 7084 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) ∈ β„•0)
27263expa 1118 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) ∈ β„•0)
2827fmpttd 7111 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0)
29 eqidd 2733 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
30 simpr 485 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ 𝑦 = 𝑖)
3130fveq2d 6892 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘–))
3231fveq2d 6892 . . . . . . . 8 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ (π‘β€˜(π‘β€˜π‘¦)) = (π‘β€˜(π‘β€˜π‘–)))
33 simpr 485 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ 𝑖 ∈ (1...𝐾))
34 fvexd 6903 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘–)) ∈ V)
3529, 32, 33, 34fvmptd 7002 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = (π‘β€˜(π‘β€˜π‘–)))
3635sumeq2dv 15645 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)))
37 fveq2 6888 . . . . . . . . 9 (𝑠 = (π‘β€˜π‘–) β†’ (π‘β€˜π‘ ) = (π‘β€˜(π‘β€˜π‘–)))
38 fzfi 13933 . . . . . . . . . 10 (1...𝐾) ∈ Fin
3938a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (1...𝐾) ∈ Fin)
4018adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
41 eqidd 2733 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘β€˜π‘–) = (π‘β€˜π‘–))
42 nn0sscn 12473 . . . . . . . . . . . 12 β„•0 βŠ† β„‚
4342a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ β„•0 βŠ† β„‚)
44 fss 6731 . . . . . . . . . . 11 ((𝑏:π‘†βŸΆβ„•0 ∧ β„•0 βŠ† β„‚) β†’ 𝑏:π‘†βŸΆβ„‚)
4515, 43, 44syl2anc 584 . . . . . . . . . 10 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏:π‘†βŸΆβ„‚)
4645ffvelcdmda 7083 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑠 ∈ 𝑆) β†’ (π‘β€˜π‘ ) ∈ β„‚)
4737, 39, 40, 41, 46fsumf1o 15665 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)))
4847eqcomd 2738 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)) = Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ))
49 fveq2 6888 . . . . . . . . . 10 (𝑠 = 𝑖 β†’ (π‘β€˜π‘ ) = (π‘β€˜π‘–))
5049cbvsumv 15638 . . . . . . . . 9 Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–)
5150a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–))
5214simprd 496 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁)
5351, 52eqtrd 2772 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = 𝑁)
5448, 53eqtrd 2772 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)) = 𝑁)
5536, 54eqtrd 2772 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁)
5628, 55jca 512 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁))
57 fzfid 13934 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (1...𝐾) ∈ Fin)
5857mptexd 7222 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ V)
59 feq1 6695 . . . . . . 7 (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) β†’ (𝑔:(1...𝐾)βŸΆβ„•0 ↔ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0))
60 simpl 483 . . . . . . . . . 10 ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∧ 𝑖 ∈ (1...𝐾)) β†’ 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
6160fveq1d 6890 . . . . . . . . 9 ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘”β€˜π‘–) = ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–))
6261sumeq2dv 15645 . . . . . . . 8 (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) β†’ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–))
6362eqeq1d 2734 . . . . . . 7 (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) β†’ (Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁))
6459, 63anbi12d 631 . . . . . 6 (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) β†’ ((𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁) ↔ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁)))
6564elabg 3665 . . . . 5 ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ V β†’ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁)))
6658, 65syl 17 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁)))
6756, 66mpbird 256 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)})
68 sticksstones17.3 . . . 4 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)}
6968a1i 11 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)})
7067, 69eleqtrrd 2836 . 2 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ 𝐴)
71 sticksstones17.6 . 2 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
7270, 71fmptd 7110 1 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  Vcvv 3474   βŠ† wss 3947   ↦ cmpt 5230  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  Fincfn 8935  β„‚cc 11104  1c1 11107  β„•0cn0 12468  ...cfz 13480  Ξ£csu 15628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629
This theorem is referenced by:  sticksstones19  40969
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