Theorem divsmulw 28135

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28159, we can eliminate the existence hypothesis (see divsmul 28162). (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
divsmulw	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem divsmulw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divsval 28131	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → (𝐴 /su 𝐶) = (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴))
2	1	eqeq1d 2735	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
3	2	3expb 1120	. . . 4 ⊢ ((𝐴 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
4	3	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
5	4	adantr 480	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
6		simpl2 1193	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → 𝐵 ∈ No )
7		simp3l 1202	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ∈ No )
8		simp3r 1203	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ≠ 0s )
9		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐴 ∈ No )
10	7, 8, 9	3jca 1128	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → (𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ))
11		noreceuw 28133	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
12	10, 11	sylan 580	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
13		oveq2 7362	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 ·s 𝑦) = (𝐶 ·s 𝐵))
14	13	eqeq1d 2735	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐶 ·s 𝑦) = 𝐴 ↔ (𝐶 ·s 𝐵) = 𝐴))
15	14	riota2 7336	. . 3 ⊢ ((𝐵 ∈ No ∧ ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
16	6, 12, 15	syl2anc 584	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
17	5, 16	bitr4d 282	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057  ∃!wreu 3345  ℩crio 7310  (class class class)co 7354   No csur 27581   0_s c0s 27769   1_s c1s 27770   ·_s cmuls 28048   /_su cdivs 28129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-1o 8393  df-2o 8394  df-nadd 8589  df-no 27584  df-slt 27585  df-bday 27586  df-sle 27687  df-sslt 27724  df-scut 27726  df-0s 27771  df-1s 27772  df-made 27791  df-old 27792  df-left 27794  df-right 27795  df-norec 27884  df-norec2 27895  df-adds 27906  df-negs 27966  df-subs 27967  df-muls 28049  df-divs 28130

This theorem is referenced by:  divsmulwd  28136  divs1  28146  divsmul  28162