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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasabl | Structured version Visualization version GIF version |
Description: A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasabl | ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harcl 9597 | . . . . 5 ⊢ (har‘𝑆) ∈ On | |
2 | onenon 9987 | . . . . 5 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (har‘𝑆) ∈ dom card |
4 | unnum 10235 | . . . 4 ⊢ ((𝑆 ∈ dom card ∧ (har‘𝑆) ∈ dom card) → (𝑆 ∪ (har‘𝑆)) ∈ dom card) | |
5 | 3, 4 | mpan2 691 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ dom card) |
6 | ssun2 4189 | . . . 4 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) | |
7 | harn0 43091 | . . . 4 ⊢ (𝑆 ∈ dom card → (har‘𝑆) ≠ ∅) | |
8 | ssn0 4410 | . . . 4 ⊢ (((har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) ∧ (har‘𝑆) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ≠ ∅) | |
9 | 6, 7, 8 | sylancr 587 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ≠ ∅) |
10 | isnumbasgrplem3 43094 | . . 3 ⊢ (((𝑆 ∪ (har‘𝑆)) ∈ dom card ∧ (𝑆 ∪ (har‘𝑆)) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) | |
11 | 5, 9, 10 | syl2anc 584 | . 2 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
12 | ablgrp 19818 | . . . . . 6 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
13 | 12 | ssriv 3999 | . . . . 5 ⊢ Abel ⊆ Grp |
14 | imass2 6123 | . . . . 5 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
16 | 15 | sseli 3991 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
17 | isnumbasgrplem2 43093 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → 𝑆 ∈ dom card) |
19 | 11, 18 | impbii 209 | 1 ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 ≠ wne 2938 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 dom cdm 5689 “ cima 5692 Oncon0 6386 ‘cfv 6563 harchar 9594 cardccrd 9973 Basecbs 17245 Grpcgrp 18964 Abelcabl 19814 |
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-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
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-nel 3045 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-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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-har 9595 df-wdom 9603 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-hash 14367 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-gic 19291 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: isnumbasgrp 43096 |
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