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Theorem sticksstones17 41567
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 4038 . . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}))
54imp 406 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)})
6 vex 3473 . . . . . . . . . . . . 13 𝑏 ∈ V
7 feq1 6697 . . . . . . . . . . . . . 14 (β„Ž = 𝑏 β†’ (β„Ž:π‘†βŸΆβ„•0 ↔ 𝑏:π‘†βŸΆβ„•0))
8 simpl 482 . . . . . . . . . . . . . . . . 17 ((β„Ž = 𝑏 ∧ 𝑖 ∈ 𝑆) β†’ β„Ž = 𝑏)
98fveq1d 6893 . . . . . . . . . . . . . . . 16 ((β„Ž = 𝑏 ∧ 𝑖 ∈ 𝑆) β†’ (β„Žβ€˜π‘–) = (π‘β€˜π‘–))
109sumeq2dv 15673 . . . . . . . . . . . . . . 15 (β„Ž = 𝑏 β†’ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–))
1110eqeq1d 2729 . . . . . . . . . . . . . 14 (β„Ž = 𝑏 β†’ (Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁))
127, 11anbi12d 630 . . . . . . . . . . . . 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 3665 . . . . . . . . . . . 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 494 . . . . . . . . . 10 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏:π‘†βŸΆβ„•0)
1615adantr 480 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑏:π‘†βŸΆβ„•0)
17163impa 1108 . . . . . . . 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 6833 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
2221adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
23223impa 1108 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
24 simp3 1136 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑦 ∈ (1...𝐾))
2523, 24ffvelcdmd 7089 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜π‘¦) ∈ 𝑆)
2617, 25ffvelcdmd 7089 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡 ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) ∈ β„•0)
27263expa 1116 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) ∈ β„•0)
2827fmpttd 7119 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))):(1...𝐾)βŸΆβ„•0)
29 eqidd 2728 . . . . . . . 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 484 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ 𝑦 = 𝑖)
3130fveq2d 6895 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘–))
3231fveq2d 6895 . . . . . . . 8 ((((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) β†’ (π‘β€˜(π‘β€˜π‘¦)) = (π‘β€˜(π‘β€˜π‘–)))
33 simpr 484 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ 𝑖 ∈ (1...𝐾))
34 fvexd 6906 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘–)) ∈ V)
3529, 32, 33, 34fvmptd 7006 . . . . . . 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 15673 . . . . . 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 6891 . . . . . . . . 9 (𝑠 = (π‘β€˜π‘–) β†’ (π‘β€˜π‘ ) = (π‘β€˜(π‘β€˜π‘–)))
38 fzfi 13961 . . . . . . . . . 10 (1...𝐾) ∈ Fin
3938a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (1...𝐾) ∈ Fin)
4018adantr 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
41 eqidd 2728 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘β€˜π‘–) = (π‘β€˜π‘–))
42 nn0sscn 12499 . . . . . . . . . . . 12 β„•0 βŠ† β„‚
4342a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ β„•0 βŠ† β„‚)
44 fss 6733 . . . . . . . . . . 11 ((𝑏:π‘†βŸΆβ„•0 ∧ β„•0 βŠ† β„‚) β†’ 𝑏:π‘†βŸΆβ„‚)
4515, 43, 44syl2anc 583 . . . . . . . . . 10 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏:π‘†βŸΆβ„‚)
4645ffvelcdmda 7088 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ 𝑠 ∈ 𝑆) β†’ (π‘β€˜π‘ ) ∈ β„‚)
4737, 39, 40, 41, 46fsumf1o 15693 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)))
4847eqcomd 2733 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)) = Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ))
49 fveq2 6891 . . . . . . . . . 10 (𝑠 = 𝑖 β†’ (π‘β€˜π‘ ) = (π‘β€˜π‘–))
5049cbvsumv 15666 . . . . . . . . 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 495 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ 𝑆 (π‘β€˜π‘–) = 𝑁)
5351, 52eqtrd 2767 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑠 ∈ 𝑆 (π‘β€˜π‘ ) = 𝑁)
5448, 53eqtrd 2767 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)(π‘β€˜(π‘β€˜π‘–)) = 𝑁)
5536, 54eqtrd 2767 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))β€˜π‘–) = 𝑁)
5628, 55jca 511 . . . 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 13962 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (1...𝐾) ∈ Fin)
5857mptexd 7230 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∈ V)
59 feq1 6697 . . . . . . 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 482 . . . . . . . . . 10 ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) ∧ 𝑖 ∈ (1...𝐾)) β†’ 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
6160fveq1d 6893 . . . . . . . . 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 15673 . . . . . . . 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 2729 . . . . . . 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 630 . . . . . 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 3663 . . . . 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 257 . . 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 2831 . 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 7118 1 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {cab 2704  Vcvv 3469   βŠ† wss 3944   ↦ cmpt 5225  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  Fincfn 8955  β„‚cc 11128  1c1 11131  β„•0cn0 12494  ...cfz 13508  Ξ£csu 15656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657
This theorem is referenced by:  sticksstones19  41569
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